Local and Multidimensional Theories of Tournaments

Thesis submitted for the degree of Doctor of Philosophy By

Avraham Morgenstern

Submitted to the Senate of the Hebrew University of Jerusalem July 2014

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This work was carried out under the supervision of:

Professor Nati Linial.

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Acknowledgement

I would like to thank my advisor professor Nati Linial for teaching and guiding me and for taking care of all my academic needs during the past 9 years.

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Abstract A tournament is an orientation of a complete graph. The study of tournaments is an important branch of graph theory and already in the 60's a whole monograph [1] was dedicated to this subject.

Tournaments are deep and in-

teresting combinatorial objects in their own right, with applications to other mathematical elds and to computer science. Still, much more remains to be discovered. Also, it is often the case that analogous problems for tournaments and for graphs are related to each other in mysterious and fascinating ways. Various basic problems are easier for tournaments than their graph analogs.

k−sets for k−cliques plus

Consider, for example, the least possible number of homogeneous constant

k.

For graphs we seek to minimize the number of

k−anti-cliques.

In tournaments we deal with transitive

k−sets,

and it is easy

to see that random tournaments are asymptotically optimal in this sense. For

k=3

Goodman established a similar phenomenon in graphs, which led Erd®s

to conjecture that this is the case for graphs with arbitrary conjecture was disproved for all

k≥4

k.

However, this

[2, 3].

In other cases the tournament problem and the graph problem are equivalent, e.g., the Erd®s-Hajnal conjecture [4]. Sometimes we just fail to establish the right connection, though it is very suggestive that a strong connection exists. Take, for example, the bounds on Ramsey numbers. It is well known that

1 2 log n, whereas in random graphs the largest cliques and anti cliques have cardinality 2 log n. every

The

n-vertex

graph contains a clique or an anti-clique of size

Erd®s-Moser theorem

says that every n-vertex tournament has a transitve log n vertices, while in random tournaments the largest tran2 log n vertices. Despite the similarity between the graph and

subtournament on sitive subset has

the tournament phenomena, no formal connection is known. The rst chapter of this thesis is a contribution to the study of high-dimensional tournaments. High-dimensional analogs of graphs - hypergraphs and simplicial complexes are the focus of intense research. Yet, very little is known about high dimensional tournaments. We consider various denitions for the notion of an

acyclic

tournament, and study their interrelations. We prove high-dimensional

counterparts for several well-known theorems: (i) The asymptotic number of n

acyclic

d-dimensional n-vertex

Moser theorem.

tournaments is

In the second chapter we study the

k -local

and (ii) The Erd®s-

As an application we generalize an argument of Erd®s and

Hajnal in Ramsey theory. The

(Θ(n))( d ) ,

prole of a graph

isomorphism types of

k -vertex

G

local proles of graphs and tournaments.

is a vector whose coordinates correspond to

graphs. The

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H -entry

is the probability that the

induced subgraph on a randomly chosen set of

Local Graph Theory

k

vertices is isomorphic to

H.

seeks to (i) Characterize the local proles of large graphs,

and (ii) Investigate global properties that can be inferred from a large graph's local prole. We show that some weak form of pseudo-randomness is implied by certain

k -universal

conditions on the local prole. A graph is called the graphs on

k -vertices

if it contains all

as induced subgraphs, i.e., all the coordinates of its

k -prole are positive. The Erd®s-Hajnal conjecture posits that every Ω (1) order n with no homogeneous subset of size n k is k-universal.

graph of We seek

local versions of this conjecture. Thus, we replace the bound on the largest homogenenous subset by a bound on the

nubmer

of homogeneous subsets of a

3 0.159... is 3-universal. Likewise every large tournament in which the fraction of size 4 transitive subsets 1 is less than 2 is 4-universal. 4 In chapter three we return to study vectors in R that are realizable as 4constant size. We prove that every large graph in which the fraction of size

cliques and that of size

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anti cliques are smaller than

proles of large tournaments. We focus on the projection to two coordinates: the probability of the transitive tournament (unique) strongly connected

4-vertex

T4

and the probability of

tournament.

C4

- the

As we show, this is equiv-

alent to several other problems: (i) Given the number of cyclic triangles in a tournament, minimize the number of copies of

C4 .

transitive triples, minimize the number of copies of

(ii) Given the number of

T4 .

(iii) Consider the edge

X e the probability that a randomly chosen vertex forms together with e a cyclic triple. For given E(X) (i.e., given the fraction of cyclic triples), we seek the minimum of V ar(X). set of a tournament as a probability space with uniform distribution, and let

be the random variable that assigns to an edge

In particular, we prove that the only tournaments in which cyclic triples are uniformly distributed among the edges, are the transitive tournament (where

1 4 − o(1)). We formulate a conjecture for all these problems, and prove bounds that

this probability vanishes) and quasi-random tournaments (where it is

almost match the conjectured values. To this end we employ the method of ag algebras.

Many open questions about the structure of large tournaments are

raised as well.

References [1] J. Moon, Topics on tournaments. Holt, Rinehart and Winston, New York-Montreal, Que.-London 1968 viii+104 pp. [2] A. Thomason, A disproof of a conjecture of Erd®s in Ramsey theory, J. London Math. Soc. (2), 39(2) (1989), 246255. [3] A. Thomason. Graph products and monochromatic multiplicities. Combinatorica, 17(1):125134, 1997.

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[4] N. Alon, J. Pach, and J. Solymosi, Ramsey-type theorems with forbidden subgraphs. Combinatorica 21 (2001), 155-170.

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Contents •

On high-dimensional acyclic tournaments

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Graphs with few 3-cliques and 3-anticliques are 3-universal

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•

On high-dimensional acyclic tournaments

38 38

•

Hebrew abstract

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Manuscripts status •

On high-dimensional acyclic tournaments - published in Discrete & Computational Geometry, Volume 50, Issue 4 (2013), Page 10851100.

Graphs with few 3-cliques and 3-anticliques are 3-universal - accepted for publication in Journal of Graph Theory. • On high-dimensional acyclic tournaments - sent to Journal of Graph Theory, still awaiting decision. •

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On high-dimensional acyclic tournaments Nati Linial∗

Avraham Morgenstern† July 11, 2013 Abstract

We study a high-dimensional analog for the notion of an acyclic (aka transitive) tournament. We give upper and lower bounds on the number of d-dimensional n-vertex acyclic tournaments. In addition, we prove that every n-vertex d-dimensional tournament contains an acyclic subtournament of Ω(log1/d n) vertices and the bound is tight. This statement for tournaments (i.e., the case d = 1) is a well-known fact. We indicate a connection between acyclic high-dimensional tournaments and Ramsey numbers of hypergraphs. We investigate as well the inter-relations among various other notions of acyclicity in high-dimensional tournaments. These include combinatorial, geometric and topological concepts.

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Introduction

A tournament is an orientation of a complete graph. The study of tournaments is a classical topic in combinatorics. Already in the 1960’s a whole monograph [11] was dedicated to this subject. Many theorems have been proved about tournaments over the years. Here we take a geometric perspective of the subject and view a tournament as an orientation of the onedimensional skeleton of a simplex. As it turns out, higher-dimensional analogs where we orient the higher skeletons of the simplex are rich in structure and raise many intriguing problems. To make the distinction clear, we often refer henceforth to traditional tournaments as 1-tournaments and to their d-dimensional counterparts as d-tournaments. As far as we know, the first paper on higher dimensional tournaments is due to Leader and Tan [9]. It is well-known and easy to prove that in a 1-tournament at most one quarter of the triples are cyclic, and they investigate higher-dimensional analogs of this statement. We start with some definitions and background material. Unless otherwise stated, every tournament that we consider has vertex set V = [n] = {1, . . . , n} with the natural order. Maintaining the topological terminology, we refer to a subset A ⊆ V as a face of dimension ∗

School of Computer Science and engineering, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Email: [email protected]. Research supported in part by the Israel Science Foundation and by a USA-Israel BSF grant. † Einstein Institute of mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Email: [email protected] 9

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Figure 1: A 2-tournament on four vertices |A| − 1. A face of dimension d is called a d-face for short, or even just a face when the relevant dimension is clear from the context. A d-tournament T = (V, ) on vertex set V is specified

V V by a mapping  : d+1 → {−1, 1}. For a d-face σ ∈ d+1 we call (σ) the orientation of σ. We mostly write σ rather than (σ). For faces τ ⊂ σ of dimension d − 1, d, respectively, define (τ ; σ) as the orientation induced on τ by the positive orientation of σ (viewed as a d-dimensional simplex). Namely, let σ = i0 < i1 < . . . < id , τ = σ \ {ij }, then (τ ; σ) = (−1)d−j . Now if σ is oriented, with orientation σ , the orientation induced on τ is σ · (τ ; σ). If τ 6⊂ σ we define (τ ; σ) to be zero.

n The incidence matrix of a d-tournament T is an nd × d+1 matrix A whose rows and columns correspond to all subsets of V of cardinality d resp. d + 1. The (τ, σ) entry of A is the orientation induced from σ to τ (and zero if τ 6⊂ σ). Clearly, T = (V, ) can be read off the incidence matrix A. We often do not distinguish between a face and the corresponding (column) vector of the incidence matrix. Note that for d = 1 these definitions yield the traditional definitions of a tournament and its incidence matrix. In order to deal with partial tournaments we allow  to take the value 0 as well. In that case, if σ = 0, the σ-column of the incidence matrix is an all-0 column. Example 1.1. Consider the 2-dimensional tournament T = ([4], (123 , 124 , 134 , 234 )) = ([4], (1, −1, 1, −1)). Figure 1 demonstrates the orientation of each of the four faces. The

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corresponding incidence matrix is 123 12 1  13 −1  14 0  23 1  24 0 34 0 

124 −1 0 1 0 −1 0

134 0 1 −1 0 0 1

234  0 0    0   −1   1  −1

Let T = (V, ) be a d-tournament with incidence matrix A and let x ∈ V . The link of x in T denoted lkT (x) is a (d − 1)-tournament on vertex set V \ {x}. It assigns to a (d − 1)-face τ the orientation that is induced on τ by τ ∪ {x} in T , namely, τ ∪{x} · (τ ; τ ∪ {x}). The degree sequence of the tournament is the vector A · ~1 where ~1 is the vector of 1’s
n of length d+1 . (Note that this definition deviates a little from the standard 1-dimensional definition. When d = 1, the x-entry of this vector is s+ (x) − s− (x), where s± (x) is the number of outgoing/incoming edges for vertex x). The degree sequence is a sequence of integers in the range [−(n − d), n − d], all of which have the same parity as n − d. A non-empty collection C of faces in a d-tournament T is called a cycle if there is a real vector v with nonnegative entries such that Av = 0, whose support (i.e., the index set of the positive coordinates in v) coincides with C. In other words there are positive real number vF for every F ∈ C s.t. X vF · F = 0. (1) F ∈C

where we identify a face with the corresponding column of the incidence matrix. E.g. The set {123, 124, 134, 234} is a cycle in the tournament T of Example 1.1 (all coefficients equal 1). Transitive (=acyclic) 1-tournaments and subtournaments are thoroughly studied, and here is a d-dimensional counterpart of this notion: Definition 1.2. A tournament T is acyclic or cycle-free if it contains no cycles. Clearly, there are exactly n! acyclic 1-tournaments on n vertices. In Section 2 we study d the number of n-vertex acyclic d-tournaments and show (Theorem 2.1) that it is nΘ(n ) . The proof(s) involve both analytic and geometric ideas. In particular, it is easy to tell from the degree vector of a 1-tournament whether or not the tournament is acyclic. As we show (Lemma 2.4) it is possible to decide whether a d-tournament is acyclic by observing its degree sequence. Let T a partial d-tournament. Using a term from the topology of simplicial complexes, a (d − 1)-face F is called free if all the d-faces that contain it induce the same orientation on F . (I.e., the corresponding row in T ’s incidence matrix is either non-negative or non-positive). An elementary collapse is a step in which we pick a free (d − 1)-face F and remove from T all 11

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the d-faces containing it, i.e., we set G = 0 for all G ⊃ F . We call T collapsible if it is possible to arrive at  = ~0 in a series of elementary collapses. Note, e.g., that the 2-tournament of Example 1.1 has no free faces and is, therefore, not collapsible. In contrast, Example 1.3. The incidence matrix of the 2-tournament T = ([4], (1, 1, 1, 1)) is 123 12 1  13 −1  14 0  23 1  24 0 34 0 

124 1 0 −1 0 1 0

134 0 1 −1 0 0 1

234  0 0    0   1   −1  1

It is easily verified that T is collapsible. Note, e.g., that the face 13 is not free, but becomes free once we collapse, e.g., the face 12. It is easy to see that a collapsible tournament must be acyclic. No subface of a face that participates in a cycle can be free. This remains so even following any sequence of elementary collapses. Let us recall the following well-known fact about 1-tournaments [6]: Theorem 1.4. Every 1-tournament on n vertices has an acyclic subtournament on log2 n vertices. There exist 1-tournaments with no acyclic subtournament on (2+o(1)) log2 n vertices. This, in particular, holds for random 1-tournaments. In Section 3 we derive a d-dimensional analog of this theorem. We show that every nvertex d-tournament has an acyclic subtournament on Ω(log1/d n) vertices and the bound is tight. There are several simple conditions on 1-tournaments which are all equivalent to “transitivity”. Namely, (i) T contains no cyclic triangles, (ii) T contains no (graph-theoretical) directed cycle, (iii) T is acyclic as defined above, (iv) T is collapsible, and finally, (v) all edges of T go forward relative to some total order on the vertices. In other words, the vertices can be mapped to R with all edges going from left to right. In Section 4 we study the implications among these notions in high-dimensional tournaments and we observe that for d > 1, the implications (v) =⇒ (iv) =⇒ (iii) =⇒ (ii) =⇒ (i) hold. We construct examples which show that all the reverse implications do not hold. This paper raises many open questions, and in Section 5 we describe a few additional directions for further research.

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1.1

Hyperplane Arrangements

A hyperplane arrangement A in Rn , or simply an arrangement is a set of hyperplanes in Rn . A chamber of A is a connected component of Rn \ (∪H∈A H). The braid arrangement is a famous example which is of relevance to us. Its hyperplanes are Hij = {x ∈ Rn | xi − xj = 0} for 1 ≤ i < j ≤ n. Its relevance to our discussion comes from the simple bijection between the chambers of the braid arrangement and permutations in Sn , or, what is the same, acyclic n-vertex 1-tournaments. For a comprehensive survey of arrangements, see [12]. As we observe below, there is a natural bijection between n-vertex acyclic d-tournaments n and the chambers of a certain arrangement in R( d ) . This arrangement may be of independent interest for other reasons as well, as we explain in Section 5.

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Enumerating acyclic tournaments

We denote by ad (n) the number of acyclic n-vertex d-tournaments. Theorem 2.1. For every integer d ≥ 1 and every large enough n there holds  ad (n) ≤

(nd) e n . d+1

Also  ad (n) ≥ where Hd is the harmonic sum Hd =

d P k=1

eHd

1 . k

 ad (n) ≥

(nd) n + on (1)

In particular, for large d,

(nd) e−γ + od (1) ·n d

where γ = 0.577 . . . is the Euler constant. Proof. We start with the lower bound which is a consequence of the following inequality on ad (n). This inequality ties between the cycles of a tournament, and the cycles of its (vertex) links. ∀n≥d≥2

ad (n) ≥

n−1 Y

ad−1 (k).

(2)

k=d−1

Before we prove Inequality (2), we use it to derive the lower bound in Theorem 2.1. For Q d = 1 the lower bound follows from Stirling’s formula, since a1 (n) = n! = n1 k. We proceed 13

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to larger d’s. The inequality implies that a2 (n) ≥

n−1 Q

k n−k , and more generally, by induction,

k=1

that ad (n) ≥

n−d+1 Y

n−k

k ( d−1 ) ,

k=1

In particular, ad (n) ≥ n

−O(nd−1 )

n Y

n−k

k ( d−1 ) .

k=1 d−1 )

By sweeping more of the error terms into the expression nO(n −O(nd−1 )

ad (n) ≥ n

n Y

n−k+d−2 d−1 k ( d−1 ) ≥ n−O(n )

n Y

, we can further write d−1 /(d−1)!

k (n−k)

.

k=1

k=1

(In the first inequality we gave up a factor of Consequently,

Qn (n−k+d−2)−(n−k) Qn O(nd−2 ) d−1 d−1 d−1 ≤ ≤ nO(n ) .) 1 k 1 k

n X (n − k)d−1

log k − O(nd−1 log n) (d − 1)! 1 Zn 1 ≥ (n − x)d−1 log x dx − O(nd−1 log n). (d − 1)!

log(ad (n)) ≥

1

The integral estimate for the sum follows from the fact that the error term is as large as the maximum of the function over the range of integration. Using the binomial formula, Z

d−1

(n − x)

 d−1  X d−1 log x 1 log x dx = (−1)r nd−1−r xr+1 ( − ). r r + 1 (r + 1)2 r=0

This gives  d−1  X log(ad (n)) d−1 log n log n 1
≥d (−1)r ( − ) − O( ). n 2 r r + 1 (r + 1) n d r=0 It only remains to verify the simple identities  d−1  X d−1 1 1 (−1)r = r r+1 d r=0 and 14

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 d−1  X d−1 1 1 1 1 1 (−1)r = (1 + + + . . . + ). 2 r (r + 1) d 2 3 d r=0 We now turn to prove Inequality (2). Proposition 2.2. Let T be a d-tournament. Suppose that for every vertex n ≥ i ≥ 1 the (d − 1)-tournament lkT (i) is acyclic, then T is acyclic. Moreover, the same conclusion holds even if we only assume that the restriction of the link lkT (i) to {i + 1, i + 2, . . . , n} is acyclic. Proof. We only state the proof of the second, stronger part of the proposition. Assume to the contrary that T contains a cycle C and X vF F = 0 (3) F ∈C

with vF > 0 for all F ∈ C (see Equation (1)). Now let n ≥ k ≥ 1 be the lowest index of a vertex in ∪F ∈C F . Let D := {F | k ∈ F ∈ C}. For F ∈ D we let F 0 := F \ {k}, and P we claim that F ∈D vF F 0 = 0, contrary to our assumption that the restriction of lkT (k) to {k+1, k+2, . . . , n} is acyclic. To see this, note that B, the incidence matrix of lkT (k)|{k+1,··· ,n} , is, possibly with a global sign reversal, a submatrix of T ’s incidence matrix A. We write A in block form as follows k∈ 6 F k 6∈ τ X1 k ∈ τ ; min(τ ) < k  0 min(τ ) = k 0 

k ∈ F ; min(F ) < k X2 X4 X6

min(F ) = k  X3  X5 X7

where the rows are indexed by (d−1)-faces τ , and the columns are indexed by d-faces F . Note that v may be viewed as a vector in the right kernel of A. It can be expressed in corresponding block form as v = (v1 , 0, v2 ), where v2 6= 0 by definition of k. It follows that v2 is in the right kernel of X7 which proves our claim, since X7 = ±B. We are now ready to complete the proof of Inequality (2), by providing a scheme that yields many acyclic d-tournaments T on vertex set [n]. Select first an arbitrary acyclic (d − 1)tournament on vertex set [2, n] to be lkT (1). Then an acyclic (d − 1)-tournament on vertex set [3, n] to be lkT (2)|[3,n] etc. By Proposition 2.2 the resulting d-tournament T is indeed acyclic. The desired inequality follows. This concludes the proof of the lower bound and we now turn to prove the upper bound. We first note that acyclic d-tournaments are uniquely determined by a their degree sequence. This simple observation gives an upper bound that is weaker than what is stated in the theorem. We still find it worthwhile to state, since several interesting questions arise in this context. 15

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Theorem 2.3. n

ad (n) ≤ n( d ) . Proof. The degree sequence of a d-tournament is an

n d




-vector whose entries have the parity n of n − d and reside in [−(n − d), n − d]. There are (n − d + 1)( d ) such vectors. The following lemma completes the proof.

Lemma 2.4. An acyclic d-tournament is uniquely determined by its degree sequence. Proof. Let T and S be two acyclic d-tournaments with respective incidence matrices A, B s.t. n A · ~1 = B · ~1. Define a vector v ∈ R(d+1) as follows ( 0, A∗,c = B∗,c vc = 1, A∗,c = −B∗,c Then A · v = 21 (A − B) · v = 12 (A − B) · ~1 = 0, so that either A = B or v 6= 0 and we found a cycle in T . This discussion raises several interesting questions concerning degree sequences. In particular we can ask • How many distinct degree sequences there are to d-dimensional n-vertex tournaments? For d = 1 quite a lot is known [8]. It would also be interesting to get some characterizations, efficient ways to recognize such sequences etc. • Of course, all acyclic n-vertex 1-tournaments have the same degree sequence, up to permutation. It seems quite intriguing to understand the degree sequences of acyclic d-dimensional tournaments for d > 1.

2.1

An improved upper bound using arrangements

In the same way that acyclic 1-tournaments are related to the braid arrangement, there are higher-dimensional counterparts to this arrangement that correspond to d-dimensional acyclic
tournaments. The arrangement in question is nd -dimensional and has one hyperplane for each d-face. The hyperplane Hσ corresponding to the d-face σ = {i0 < . . . < id } is defined by the equation d X X (τ ; σ)xτ = (−1)d−k xσ\{ik } = 0. τ

k=0

n (where the coordinates of the vectors in R( d ) are indexed by

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[n] d


).

There is a natural bijection between chambers of this arrangement and acyclic d-tournaments on vertex set [n]: Corresponding to a chamber C of the arrangement is the tournament that orients the face σ = i0 < . . . < id according to the rule d X σ = sgn( (−1)k xσ\{ik } ), k=0

where x is an arbitrary point in C. It is easy to see that the orientation does not depend on the choice of x ∈ C. In the opposite direction, we want to associate a chamber C to a given acyclic tournament T = ([n], ). Equivalently, it suffices to specify a point x ∈ C. This means that we must show the consistency of the following system of inequalities fσ (x) := σ

d X

(−1)k xσ\{ik } > 0 for every d-face σ = {i0 < . . . < id }.

k=0

By linear programming duality this system is inconsistent iff there exists a nonnegative linear combination of the fσ that is identically zero. Namely, there exist ασ ≥ 0 not all zero, s.t. P σ ασ fσ = 0. But such ασ constitute the coefficients of a cycle in T . We next recall the well-known fact (e.g., [10]) that an n-dimensional hyperplane arrangen
P m ment with m hyperplanes has at most chambers. We can now complete the proof of k k=0

the upper bound in Theorem 2.1. For n large enough, ad (n) ≤

(nd)  X k=0

n d+1




k

n n n n d+1 n−d ≤ 2(d+1)H(( d )/(d+1)) = 2 d+1 H( n−d )( d )

It is not hard to verify that for 1 ≥ x ≥ 0 the binary entropy function satisfies H(x) ≤ x · log2 ( xe ), which yields  (nd) n−d ad (n) ≤ ·e . d+1 The claim follows.

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Large acyclic subtournaments 1

Theorem 3.1. Every d-tournament on n vertices has an acyclic subtournament on Ω(log d (n)) vertices. This bound is tight up to a constant factor, and is attained, in particular, by random d-tournaments. Proof. We will go through the (d − 1)-faces in their reverse lexicographic order and eliminate some vertices along the way. Consider the current (d − 1)-face τ and the set S of all currently 17

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remaining vertices that precede all the vertices of τ . We delete some of the elements x ∈ S according to the following criterion. The d-face σ = τ ∪ {x} has its orientation σ and it induces an orientation on τ . This splits S into two parts according to the orientation induced on τ . We eliminate all the vertices in the smaller of these two parts and all (d − 1)-faces that contain an eliminated vertex. We make two claims: • The remaining tournament is collapsible, and hence acyclic. • At least Ω(log1/d n) vertices survive the whole process. To prove the first claim, note that the minimal face (in reverse lexicographic order) is free. After that face is being collapsed, the next minimal face becomes free once again, etc. For the second claim note that at each step, the size of the remaining vertex set is at least a half of its previous size. Let K be the set of vertices that survive the whole process, and
let |K| = k. The collection of (d − 1)-faces τ that are examined in the process is exactly Kd . k Consequently, n/2(d) ≤ k, which yields the claimed bound k ≥ Ω(log1/d n). Tightness follows from Theorem 2.1 combined with a simple first moment argument. Fix integers n, k and d and consider a random d-tournament T on n vertices. Let X be the random variable that counts the number of acyclic k-vertex subtournaments of T . It follows from Theorem 2.1 that   O(kd ) n k E(X) ≤ . k ) k 2(d+1 Consequently, there is a value of k ≤ O(log1/d n) for which E(X) < 1. The conclusion follows. It is well known and easy to show [6] that every n-vertex 1-tournament contains an acyclic subtournament on log2 n vertices and that in a random 1-tournament the largest acyclic subtournament has (2 + o(1)) log2 n vertices. However, despite many attempts, it seems difficult to close this gap. We therefore suspect that closing the gap between the upper and the lower bound in Theorem 3.1 will not be an easy task.

3.1

A connection with Ramsey theory

As usual, we denote by Rd (l, k) the smallest integer n for which the following holds. Every red/blue coloring of the hyperedges in the complete n-vertex d-uniform hypergraph contains either a complete l-vertex red hypergraph or a complete k-vertex blue hypergraph. Relatively little is known about the growth rate of these numbers for d > 2. In a recent paper [3], Conlon, Fox and Sudakov ask in particular, whether Rd (d + 1, k) grows like a tower of height (d − 1) 2 22

···2

k

in k (i.e., 2 ). We are unable to answer their question, but we note that the notion of acyclic d-tournaments allows us to extend an old argument of Erd˝os and Hajnal [4] and show 18

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d

Theorem 3.2. For all d ≥ 1, Rd+2 (d + 3, k) ≥ 2cd k . The smallest cyclic d-tournament has d + 2 vertices. We refer to this tournament simply as a (d + 2)-cycle and we note that on a given set of d + 2 vertices there are exactly two possible (d + 2)-cycles. In particular, the probability that a random (d + 2)-vertex d-tournament is a (d + 2)-cycle is 2−d−1 . Lemma 3.3. The probability that a random n-vertex d-tournament contains no (d + 2)-cycle d+1 is at most 2−Ω(n ) . Proof. The Erd˝os-Hanani Conjecture was proved by R¨odl (e.g., [2]). It implies the existence of a large system of (d + 2)-sets of vertices no two of which have d + 1 vertices in common. n (d+1 ) nd+1 = (1 − on (1)) (d+2)! sets. As noted Specifically, there exists such a system of (1 − on (1)) d+2 −d−1 above, each member of this system is a cycle with probability 2 , and the claim follows, since these events are independent. 1

Corollary 3.4. There exist n-vertex d-tournaments in which every subtournament on c0d log d n vertices contains a (d + 2)-cycle. This in particular holds with positive probability for random d-tournaments. Here c0d > 0 is a constant that depends only on d. Proof. The claim follows from a first-moment argument. Let T be a random n-vertex dtournament, and let X be the number of k-vertex subtournaments of T that contain no (d + 2)-cycle. By the previous lemma   n −Ω(kd+1 ) E(X) ≤ 2 k 1

For k = c0d log d n this expectation is less than 1 and the claim follows. We can complete now the proof of Theorem 3.2. A d-tournament as in Corollary 3.4,
[n] induces a red/blue coloring of the subsets U ∈ d+2 as follows. We color U blue if it is a (d + 2)-cycle, and red otherwise. The claim follows since no set of d + 3 vertices is entirely blue. To see this, let S be a set of d vertices where x1 , x2 , x3 are the three remaining vertices. Consider the orientation induced on S by each of the three faces S ∪ {xi } for i = 1, 2, 3. There must be two of these orientations, say for i = 1, 2 for which the orientations on S coincide. But then S ∪ {x1 , x2 } is not a (d + 2)-cycle.

4

Alternative notions of acyclicity

We find it instructive to recall now the one-dimensional situation and see how things change as the dimension grows. Indeed all of the following properties of a 1-tournament T are easily seen to be equivalent. 19

11

1. T contains no cyclic triangles. 2. T contains no (graph-theoretical) directed cycle. 3. T is acyclic as defined above. 4. T is collapsible. 5. All edges of T go forward relative to some total order on the vertices. In other words, the vertices can be mapped to R with all edges going from left to right. As noted below, all the above properties of a tournament T have d-dimensional counterparts as follows. 1. T contains no (d + 2)-cycle. 2. There is no nonempty set of faces in T that sum to zero. In other words, zero is the only solution of Equation (1) in 0/1 coefficients. 3. T is acyclic. 4. T is collapsible. 5. Fix the positive orientation on Rd . The orientation of the faces of T is induced from some general-position embedding of its vertices in Rd . As we presently note, these conditions appear in increasing order of strength. We subsequently present examples that show that reverse implications need not hold. Proposition 4.1. The following implications among d-dimensional tournaments hold: 5⇒4⇒3⇒2⇒1 Proof. Most implications are very easy to verify and we prove here only the implication 5⇒4 (that 4⇒3 was already mentioned before). Let T be a d-tournament that is realizable by means of an embedding ι of the vertex set in Rd . Every (d − 1)-face on the boundary of the convex hull conv(image(ι)) is free. Once those are eliminated, the new boundary (d − 1)-faces are free again, etc. We turn to show that the reverse implications do not hold. We note that it suffices to consider 2-dimensional examples to this end. Proposition 4.2. 16⇒2

20

12

Proof. Consider the three-dimensional octahedron conv(±e1 , ±e2 , ±e3 ). We orient its eight triangular facets according to the outer normal of this polytope. These eight faces sum to zero, hence this 2-tournament is 0/1-cyclic. Each of the remaining twelve 2-faces contains an edge of the form [−ei , ei ] for some 3 ≥ i ≥ 1. We orient these faces so that the orientation induced on this edge is −ei → ei . Let us show that no set of four vertices can form a d + 2 = 4-cycle. Every set of four vertices must contain at least one of the pairs {−ei , ei } and cannot, therefore, be cyclic. Proposition 4.3. 26⇒3 Proof. Start with the standard 6-point triangulation of the projective plane in Figure 2, where each of the 10 faces is oriented clockwise. Now add the face σ = {1, 2, 3} with the orientation σ = +1 i.e., 1 → 2 → 3 → 1. Note that the sum of the 10 faces +2σ is zero. Hence this partial tournament is already cyclic, but contains no cycle in 0/1 coefficients. We could try and orient the remaining 9 faces so as to maintain the property that there is no cycle in 0/1 coefficients, but this plan must fail. Consider the face ρ = {3, 4, 5} and note that both ±ρ are expressible as 0/1 combinations of already oriented faces, namely ρ = σ + {1, 3, 4} + {1, 4, 5} + {1, 2, 5} + {2, 3, 5} −ρ = σ + {2, 3, 4} + {2, 4, 6} + {4, 5, 6} + {3, 5, 6} + {1, 2, 6} + {1, 3, 6}. Consequently, however we orient ρ, a 0/1-cycle is created. As it turns out, this plan does work if we slightly modify the above construction. Start instead from the 10 point triangulation in Figure 2 along with the face σ = {1, 2, 3}. Again the 18 faces in this triangulation are oriented clockwise and σ = +1. Consequently, • This partial tournament has a single cycle, not in 0/1 coefficients. • No additional cycle can be created by the addition of any single oriented face. It follows that we can orient the remaining faces one by one so as to preserve the first property. Let ρ be a face which wasn’t oriented yet. Suppose that both orientations of ρ are creating new P P 0 P cycles. Namely, vF F + ρ = 0 and vF F − ρ = 0. Then (vF + vF0 )F = 0 is another cycle, which doesn’t involve ρ. Hence, this must be (a positive constant times) the only existing cycle. This means that the two new cycles created by ±ρ are using only the original faces and ρ, contrary to the above second property. Proposition 4.4. 36⇒4 9 Proof. Using a computer search, we found a point x ∈ R(2) which doesn’t satisfy any of the equations xij −xik +xjk = 0 (i < j < k). As explained above, such a point belongs to a unique

21

13

Figure 2: The standard 6 point triangulation of the real projective plane, and a modified 10 point triangulation. chamber which corresponds to an acyclic 9-vertex 2-tournament. The resulting tournament is checked by the computer not to be collapsible. The point we found is x = (42, 0, 3, 88, 91, 87, 66, 28, 64, 60, 87, 11, 39, 81, 37, 51, 0, 23, 77, 33, 23, 58, 11, 7, 70, 64, 73, 57, 86, 52, 98, 49, 57, 100, 43, 60). (Coordinates are indexed by unordered pairs and appear in lexicographic order). 10 Better still, consider the following point in R( 2 ) . x = (76, 61, 70, 6, 95, 97, 45, 11, 26, 12, 33, 93, 5, 97, 92, 9, 48, 26, 58, 82, 4, 96, 14, 83, 87, 92, 93, 92, 92, 18, 64, 11, 76, 4, 39, 82, 24, 94, 25, 36, 30, 40, 64, 21, 7). This example is easier to verify, since the acyclic 10-vertex 2-tournament corresponding to x has no free faces. Proposition 4.5. 46⇒5 Proof. We describe a collapsible 5-vertex 2-tournament which is not realizable in the plane. We note first that there are precisely six isomorphism types of acyclic 5-vertex 2-tournaments, three of which are realizable in the plane. All other three are collapsible. We prove this for a specific tournament: Let vertices 1, 2, 3 be any triangle in the plane, with 4, 5 being mapped to the same point in the interior of the triangle (Think of 5 as residing ’above’ 4). All faces, except for the three faces containing the edge {4, 5} are orientated clockwise. These three faces are oriented consistently with the orientation 4 → 5 of the edge {4, 5}. The resulting 2-tournament is collapsible, since all the edges {4, 5}, {1, 2}, {1, 3}, {2, 3} are free, and each face contains one of these edges. 22

14

To see that this tournament is not realizable in the plane, note that its set of free edges form a disconnected graph, whereas in a realizable d-tournament the free (d − 1)-faces form a (topological) cycle. Although we are mostly concerned here with acyclic d-tournaments, we find the other notions interesting as well. Below we make some comments about them, with a special interest in the relevant enumeration problems.

4.1

Avoiding a (d + 2)-cycle

Leader and Tan’s notion of a d-dimensional cycle [9] (which they call “a directed simplex”) coincides with our (d + 2)-cycle. In Section 3.1 we make several observations concerning (d + 2)-cycle-free d-tournaments. As mentioned, it is a classical fact that the largest acyclic subtournament of a random 1-tournament has logarithmic order. Corollary 3.4 gives a ddimensional analog of this statement, which is even stronger, since “acyclic” is replaced by “contains no (d + 2)-cycle”. We give here upper bounds on the number of acyclic tournaments under the various interpretations of acyclicity, but we still do not know how many n-vertex (d + 2)-cycle-free d-tournaments there are. The proof of Lemma 3.3 gives the upper bound 2

1+o(1) n d+2 d+1

(

)

which far exceeds the other upper bounds proved here on the number of “acyclic” tournaments. Whether or not this gap is inevitable we do not know.

4.2

No 0/1-cycles

Theorem 2.3 is initially presented as an easy way to derive an upper bound on the number of acyclic tournaments. However its proof and specifically Lemma 2.4 show the equivalence of the following two classes of d-tournaments: (i) Those that contain no cycle in 0/1 coefficients, and (ii) Those that are uniquely reconstructible from their degree sequence. In particular, Theorem 2.3 gives an upper bound on the number of d-tournaments with no 0/1-cycles.

4.3

Collapsible d-tournaments

The lower bound in Theorem 2.1 actually applies to the smaller class of collapsible d-tournaments. To see this, note that inequality (2) can be proved likewise if “acyclic” is replaced by “collapsible”. It is interesting to get better upper and lower bounds on the number of collapsible n-vertex d-tournaments. In particular, better lower bounds will improve the lower bound in Theorem 2.1. 23

15

4.4

Realizable d-tournaments

An Rd -realizable d-tournaments is synonimous with an n-point order type in Rd . Order types 2 are of much interest in discrete geometry (see [10]). Their number is known to be n(1+o(1))d n [7].

5

Some final comments and open questions

We feel compelled to recall the following well-known conjecture Conjecture 5.1 (Erd˝os-Hajnal [5]). For every graph H there is a γ > 0 such that every n-vertex graph which contains no induced copy of H must have either a clique or an anticlique of cardinality ≥ nγ . As shown in [1] this conjecture can be restated as follows: Conjecture 5.2. For every 1-tournament F there is a γ > 0 such that every n-vertex 1tournament which contains no copy of F has an acyclic subtournament on ≥ nγ vertices. It would be interesting to consider high-dimensional analogs of these statements. n In Section 2.1 we study a hyperplane arrangement in R( d ) whose cells are in 1 : 1 correspondence with the acyclic n-vertex d-tournaments. It is also interesting to consider the n analogous arrangements in C( d ) and in particular ask about the fundamental group of these complex arrangements. The situation for 1-tournaments is well understood. Namely, for 1 ≤ i < j ≤ n let Hij be the hyperplane defined by the equation xi = xj . The fundamental group π1 (Cn \ ∪ Hij ) is the so-called pure braid group, a mathematical object of great impori
tance and interest. It is an intriguing possibility that there are interesting groups waiting to be discovered for larger d. For example, it would be interesting to determine the fundamental n group π1 (C( 2 ) \ ∪ Hijk ), where for 1 ≤ i < j < k ≤ n we define Hijk as the hyperplane in i
n C( 2 ) whose equation is xij + xjk = xik .

6

Acknowledgement

We were not sure for a while which of the many notions of acyclicity would be of greatest interest to study. We are grateful to Roy Meshulam for helping us take the (hopefully) right decision.

References [1] N. Alon; J. Pach; J. Solymosi, Ramsey-type theorems with forbidden subgraphs. Combinatorica 21 (2001), 155-170. 24

16

[2] N. Alon; J. Spencer, The Probabilistic Method (2ed). Wiley-Interscience Series in Discrete Mathematics and Optimization, 2000. [3] D. Conlon; J. Fox; B. Sudakov, An improved bound for the stepping-up lemma. arXiv:0907.0283 [math.CO]. [4] P. Erd˝os; A. Hajnal, On Ramsey like theorems, problems and results. Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972) , pp. 123-140, Inst. Math. Appl., Southend-on-Sea, 1972. [5] P. Erd˝os; A. Hajnal, Ramsey-type theorems. Discrete Applied Mathematics 25(1989), 37-52. [6] P. Erd˝os; L. Moser, On the representation of directed graphs as unions of orderings. Publ. Math. Inst. Hungar Acad. Sci. 9, 125-132 (1964). [7] J. Goodman; R. Pollack, The complexity of point configurations. Discrete Applied Mathematics 31 (1991) 167-180. [8] D. Kleitman; K. Winston, On the asymptotic number of tournament score sequences. J. Combin. Theory Ser. A 35 (1983), no. 2, 208-230. [9] I. Leader; T. Tan, Directed simplices in higher order tournaments. Mathematika 56 (2010), no. 1, 173-181. [10] J. Matouˇsek; Lectures on discrete geometry. Graduate Texts in Mathematics, 212. Springer-Verlag, New York, 2002. xvi+481 pp. [11] J. Moon, Topics on tournaments. Holt, Rinehart and Winston, New York-Montreal, Que.-London 1968 viii+104 pp. [12] R. Stanley, An introduction to hyperplane arrangements. Geometric combinatorics, 389-496, IAS/Park City Math. Ser., 13, Amer. Math. Soc. 2007.

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17

1

Graphs with few 3-cliques and 3-anticliques are 3-universal Nati Linial∗

Avraham Morgenstern†

February 18, 2014 Abstract For given integers k, l we ask whether every large graph with a sufficiently small number of k-cliques and k-anticliques must contain an induced copy of every l-vertex graph. Here we prove this claim for k = l = 3 with a sharp bound. A similar phenomenon is established as well for tournaments with k = l = 4.

1

Introduction

We start by recalling the notion of universality. Definition 1.1. A graph (resp. tournament) is called l-universal if it contains every l-vertex graph (tournament) as an induced subgraph (subtournament). We next recall the celebrated Erd˝os-Hajnal conjecture [5] that we reformulate in a somewhat nonstandard form. As usual we denote by ω(G), α(G) the clique, resp. anticlique number of the graph G. Conjecture 1.2. [Erd˝os-Hajnal] For every integer l there is an  > 0 such that every n-vertex graph G with α(G), ω(G) < n is l-universal. The largest size of a transitive subtournament of the tournament T is denoted by tr(T ). The Erd˝os-Hajnal conjecture for tournaments states: Conjecture 1.3. For every integer l there is an  > 0 such that every n-vertex tournament T with tr(T ) < n is l-universal. ∗

School of Computer Science and engineering, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Email: [email protected]. Research supported in part by the Israel Science Foundation and by a USA-Israel BSF grant. † Einstein Institute of mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Email: [email protected]

26

1

As shown by Alon, Pach and Solymosi [2], these two conjectures are equivalent. The Erd˝os-Hajnal conjecture in both its formulations posits that a graph (resp. a tournament) which satisfies a rather mild upper bound on largest clique and anticlique (resp. transitive set) must be l-universal. In this paper we ask the following Problem 1.4. For given integers k, l is every large graph with few k-cliques and k-anticliques necessarily l-universal? Similarly, is a large tournament with only few transitive subtournaments of order k necessarily l-universal? The answer for the graph problem with k = l = 3 turns out to be positive, and we derive a sharp bound for this statement. For tournaments, the case of k = 3 is trivial, but the range k ≥ 4 turns out rather interesting. We prove that an upper bound on the number of transitive 4-vertex subtournaments implies 4-universality. As explained in the last section, this line of thought can be developed in numerous additional ways. We need some definitions and notations which we state in the language of graphs. Their counterparts for tournaments are obvious. For a fixed l-vertex graph H and an arbitrary graph G we denote by p(H, G) the probability that a randomly chosen set of l vertices in G induces a subgraph that is isomorphic to H. Given an integer l, we let Hl be the list of all N = Nl isomorphism classes of l-vertex graphs. We refer to the vector πl (G) = (p(H, G))H∈Hl as the l-th local profile of the graph G. Below we use G to always denote a sequence of graphs Gn with |V (Gn )| → ∞. If the limit limn λ(Gn ) exists, where λ is some graph parameter, we denote this limit by λ(G). Likewise ¯ λ(G) := lim supn λ(Gn ) and λ(G) := lim inf n λ(Gn ). For each i = 0, 1, 2, 3 there is exactly one graph Pi ∈ H3 that has i edges, and we denote p(Pi , G) by pi (G), or simply by pi when G is clear from the context. We note that p0 (G) (resp. p1 (G)) equals p3 (resp. p2 ) of its complement graph. For example, in our terminology, Goodman’s well-known bound [9] takes the form: Theorem 1.5. [Goodman] For every G there holds p0 + p3 (G) ≥ 14 . Jacob Fox (personal communication) has observed that the answer to Problem 1.4 is positive for some l = Ω(k). Namely, he found the following lemma whose proof appears in Section 4. k

k

Proposition 1.6. If both p(Kk , G) < 2−(2) + and p(K k , G) < 2−(2) + then G is ck-universal, where c > 0 is a universal constant. We define Πl ⊂ RN as the set of all points π ∈ RN for which there exists a sequence of graphs G with πl (G) = π. It is still a major open question to get a good description of these sets Πl . In the present article we add some piece to what is known about Π3 . At this writing even Π3 is not yet fully understood (but see [11, 16]). The state of our knowledge of Πl for l ≥ 4 is really very limited, though some work already exists, e.g., [7, 8, 12, 13, 17, 18, 19]. 27

2

Much of the recent progress in this area was achieved using Razborov’s flag algebras method. We say that G is l-universal if p(H, G) > 0 for every H ∈ Hl . Our main result is Theorem 1.7. There is a constant ρ = 0.159181... such that every G with p0 (G), p3 (G) < ρ is 3-universal. The bound is tight. The number ρ is defined as ρ = 6θ2 (1 − 2θ) where θ = 0.427373... is the largest root of θ3 + θ2 − θ + 61 = 0. We prove this theorem in Section 2. In Section 3 we state and prove our results for tournaments. In Section 4 we prove Proposition 1.6 and mention several open questions.

2

Proof of Theorem 1.7 for graphs

First, note that by Goodman’s theorem 1.5, p0 , p3 ≤ ρ <

1 =⇒ p0 , p3 > 0. 4

It remains to prove that p1 , p2 > 0. By the above-mentioned symmetry between p1 and p2 , it suffices to consider only p2 . By passing to a subsequence, if necessary, and arguing by contradiction, it suffices to consider only sequences G with p2 (G) = 0. By the graph removal lemma [1, 3], an n-vertex graph G with p2 (G) = o(1) can be made P2 -free1 by flipping only o(n2 ) edges2 . Since this changes p0 (G) and p3 (G) by only o(1), we may apply this removal step to all G ∈ G, and assume that every G ∈ G is P2 -free. But a graph is P2 -free iff it is a union of vertex disjoint cliques, so these are the only graphs we consider henceforth. Our goal is to prove that max(p0 , p3 ) ≥ ρ for such graphs. We proceed with a series of reductions which allow us to make the following assumptions: 1. There is a bound on the number of cliques in all G ∈ G. 2. Both limits p0 (G) and p3 (G) exist. 3. p0 (G) = p3 (G). Under these assumptions, the theorem follows from Lemma 2.1 below. It suffices to consider n-vertex graphs G with only a bounded number of non-trivial cliques. For let us fix some  > 0 and remove all the edges from every clique of size < n. This leaves 1

Note that P2 is a 3-vertex path. This notation is not universally accepted, but hopefully no confusion is created. 2 Here o(1) means on (1). In general, little-oh terms are taken w.r.t. to the order of the graph that tends to infinity. 28

3

only < 1 non-trivial cliques in G which now has the desired form. This changes the parameters p0 (G), p3 (G) only by O(). By letting  → 0 the reduction follows. Our next reduction is to graphs G with |p0 (G) − p3 (G)| ≤ O( n1 ). Given the additional assumption that p0 (G) exists, this will imply p0 (G) = p3 (G). Suppose that p0 (G) − p3 (G)  n1 for G an n-vertex graph which is the union of vertex-disjoint cliques. We construct another n-vertex graph G0 with p0 (G) > p0 (G0 ), p3 (G) < p3 (G0 ) and |p0 (G0 ) − p3 (G0 )| ≤ O( n1 ). This G0 is also the disjoint union of vertex-disjoint cliques and has no more cliques than G. To construct G0 we sequentially move vertices from the smallest clique3 in G to the largest one, breaking ties arbitrarily, thereby changing p0 and p3 by at most O( n1 ). We stop when |p0 − p3 | ≤ O( n1 ). The case p0 (G) < p3 (G) is similar, but even simpler. We sequentially isolate vertices until |p0 − p3 | ≤ O( n1 ). The last reduction is achieved by passing to a subsequence in which the limits p0 = p0 (G) and p3 = p3 (G) exist and are equal. By passing to a subsequence if necessary we can fix the bound r on the number of nontrivial cliques and the relative sizes α1 , α2 , . . . , αr ≥ 0 of these cliques. In other words, we can now restrict ourselves to a sequence G whose n-th graph is Gn = Kα1 n tKα2 n t. . .tKαr n tK βn P where α1 , α2 , . . . , αr ≥ 0 and β = 1 − αi ≥ 0. We ignore issues of rounding αj n to integral values since this affects the relevant parameters by only an additive O( n1 ) term 4 . The next lemma deals with graphs of this particular structure. P Lemma 2.1. Let α1 , . . . , αr ≥ 0 and β = 1 − αi ≥ 0. Let X X X X p3 = αi3 and p0 = 6 αi αj αk + 6β αi αj + 3β 2 αi + β 3 . i
i
If p0 = p3 then p0 , p3 ≥ ρ = 0.159181... This bound is tight. Proof. We apply the Lagrange multipliers method to determine the smallest possible value of P max(p0 , p3 ) under the constraints p0 = p3 , αi ≥ 0, αi ≤ 1. (We eliminate the variable β by P substituting β = 1 − αi ). There are three cases to consider: • The minimum is attained in the interior of this region. We calculate the partial deriva3 tives of the objective function ∂p = 3αl2 and the derivatives of the constraint ∂αl X X X X X ∂(p3 − p0 ) = 3αl2 − 6 αi αj − 6(1 − αi ) αi + 6 αi αj − 3(1 − αi )2 + ∂αl i i
If there is an isolated vertex in the graph, the corresponding clique gets eliminated, but this creates no problem in the argument. 4 To see this designate one vertex in each nontrivial clique as “special”. The difference between the calculations below and the exact values comes only from triples that contain a special vertex. 29

4

6(1−

X

αi )

i

X i

αi +3(1−

X

αi )2 = 3αl2 +6αl

i

X

αi +6(1−

X

αi )αl = 3αl (αl +2(1−αl )).

i6=l

The Lagrange multipliers method implies that at a critical point there holds 3 −p0 ) λ ∂(p∂α , l

∂p3 ∂αl

=

where λ is a Lagrange multiplier. Consequently αl = λ(2 − αl ) for all l (since we are working in the interior of our region, all αl are positive). This is a linear equation, so all αl are equal. If r ≥ 3, then p3 ≤ 3( 13 )3 = 91 , and p0 ≥ 14 − p3 > 0.13 (by Goodman’s theorem), hence p0 6= p3 , a contradiction. If r = 1, p3 = α13 = p0 = (1−α1 )3 +3(1−α1 )2 α1 . The solution is α1 = 0.652704..., for which p0 = p3 = 0.278... > ρ. If r = 2, p3 = 2α13 = p0 = (1 − 2α1 )3 + 3(1 − 2α1 )2 · 2α1 + 6(1 − 2α1 )α12 . The solution is α1 = 0.442125..., and p0 = p3 = 0.172848... > ρ. • The minimum is attained when αi = 0 for some i. This case is solved by removing this αi using induction on r. P • The minimum is attained when ∀i αi > 0, and αi = 1. We add the constraint P αi = 1 to our Lagrange multipliers equations. This gives αi2 = λ(2αi − αi2 ) + µ. All αi satisfy this quadratic equation, so they all take at most two different values, say α1 appears s times and α2 appears t times with sα1 + tα2 = 1. We can assume that s, t > 0, and α1 > α2 > 0 since the case of equal α’s was already treated above. If s ≥ 3 then p3 ≤ 19 , and it follows (as before), that p0 6= p3 , a contradiction.

If s = 1, p3 = α13 + tα23 , p0 = 6 2t α1 α22 + 6 3t α23 , and α1 = 1 − tα2 . Denote x = tα2 . 3 t . We have p3 = (1 − x)3 + xt2 and p0 = 3x2 − 2x3 − 3t x2 + t22 x3 . Let Here, 0 < x < t+1 τ (x) = p3 − p0 . The value of x is determined by the equation τ (x) = 0. Note that t τ (0) > 0, and τ 0 (x) < 0 for 0 < x < t+1 . Therefore, τ is decreasing, and there is a unique solution for τ (x) = 0. 1 Now, τ ( 13 ) = 27 + 3t1 − 27t1 2 > 0 implies that x > 31 . This implies that p0 (x) > p0 ( 31 ), 7 7 1 since, p00 > 0. It remains to compute, for t ≥ 4, p0 ( 13 ) > 27 − 3t1 ≥ 27 − 12 = 0.175... > ρ.

The case s = t = 1 is vacuous, since p3 > p0 = 0. If s = 1, t = 2 the equation in x = 2α2 is 1 − 3x + 32 x2 + 43 x3 = 0 with root at x = 0.469285..., and p0 = p3 = 0.1753... > ρ. If s = 1, t = 3 then x = 3α2 satisfies 1 − 3x + x2 + 89 x3 = 0 so that x = 0.409632..., and p0 = p3 = 0.2134... > ρ. This concludes the case s = 1, and the only remaining case to analyze is s = 2. Again, 3 3 x := tα2 . Here, p3 = (1−x) + xt2 and p0 = 32 x − 12 x3 − 3t x2 + t22 x3 . The range of x 4 t is 0 < x < t+2 . We first consider the case t ≥ 3. Define τ (x) = p3 (x) − p0 (x) = 1 9 3 2 1 3 t − 4 x + 4 x + 4 x + 3t x2 − t12 x3 . Again, τ (0) > 0, and τ 0 < 0 for 0 < x < t+2 , hence τ 4 0 decreases. Also, p0 increases, since p0 > 0. 30

5

Let x0 = 0.115749... be the solution in [0, 1] of the equation 41 − 94 x + 34 x2 + 14 x3 = 0. τ (x0 ) = 3t x20 − t12 x30 > 0. Since τ decreases, the solution for τ = 0 is bigger than x0 . Since p0 increases, the optimal value of p0 is larger than p0 (x0 ) = 0.172848...− 0.040193... + t 0.003102... ≥ 0.159450... > ρ. t2 If s = 2, t = 1, then p0 = 6α12 (1 − 2α1 ), and p3 = 2α13 + (1 − 2α1 )3 . Solving for p0 = p3 gives α1 = 0.234643... or α1 = 0.427373.... Since α1 > α2 = 1 − 2α1 , we have α1 = 0.427373..., and p0 = p3 = ρ. This example proves the tightness claim in the lemma. p3 = 2α13 + 2α23 = Finally, s = 2, t = 2 gives p0 = 6(2α12 α2 + 2α1 α22 ) = 3α1 (1 − 2α1 ) and √ 1 − 23 α1 + 3α12 . Solving for p0 = p3 with α1 > α2 gives α1 = 3+12 5 = 0.436338..., and 4 p0 = p3 = 16 > ρ.

3

On 4-profiles of tournaments

As in the discussion above, we consider families T of tournaments of orders going to ∞ and discuss their k-local profiles. Likewise we define the limit values sets πl (T ) and the limit sets Πl of tournaments. The 3-profiles of tournaments are easy and completely understood. There are just two 3-vertex tournaments, one transitive and one cyclic with frequencies t3 and c3 = 1 − t3 respectively. It is well-known and easy to prove that for every T there holds t3 (T ) ≥ 34 and this is all there is to 3-profiles of tournaments. In this section we prove the analog of Theorem 1.7 for tournaments and k = 4. In addition we derive some information on Π4 (tournaments). There are exactly four isomorphism types of 4-vertex tournaments, see figure 1. Their names are as follows: • T4 is the transitive 4-tournament. • C4 is the (one and only) strongly connected 4-tournament. • In W4 there is a cyclic triangle all three vertices of which arrow the fourth vertex. • In L4 one vertex arrows all the three vertices of a cyclic triangle We use the shorthand t4 (T ), c4 (T ), w4 (T ), l4 (T ) for p(T4 , T ), etc., or even do not mention T explicitly when clear from the context. Note that if the limits t4 , c4 , w4 , l4 exist for some family of tournaments T , then the limit fraction of cyclic triangles c3 (T ) exists as well and equals 2c4 +w4 4 +l4 . We recall the class C = Cn of circular tournaments of odd order n. The n vertices of Cn are equally spaced on the unit circle, with an edge x → y iff the clockwise arc from x to y is shorter than the counter clockwise arc between them. We are now ready to state our theorem for tournaments: 31

6

Figure 1: The four types of 4-vertex tournaments (in order): T4 , C4 , W4 , L4 . Theorem 3.1. Every family of tournaments T for which t4 (T ) < w4 (T ), l4 (T ) ≥ 12 − t4 (T ). Also c4 ≥ 16 when t4 (T ) ≤ 12 . The circular tournaments satisfy t4 (C) = 21 and yet l4 = w4 = 0.

1 2

is 4-universal. Moreover,

Remark 3.2. We do not know whether the inequality c4 ≥ 16 is tight, and so we ask how small c4 (T ) can be when t4 (T ) = 21 . A similar question is presented in remark 3.5. Proof. The theorem follows from the proposition below. In more detail, the positivity of t4 , l4 , and w4 follows from items (2), (3), and (4) respectively. The lower bound on c4 follows by combining (5) with the equality c3 = 1−t44+c4 . Proposition 3.3. The following inequalities hold for every (t4 , c4 , w4 , l4 ) ∈ Π4 . c4 ≤ t4

(1)

3 ≤ t4 8

(2)

1 2 1 t4 + w4 ≥ 2 All the above inequalities are tight. In addition: t4 + l4 ≥

c4 ≥ 6c23 .

(3) (4)

(5)

Remark 3.4. These inequalities, four linear and one quadratic, provide some information on the set Π4 . It would be interesting to derive a full description of Π4 . Remark 3.5. We still do not know how tight inequality (5) is and we ask how small c4 (T ) can be, given c3 (T ). It is not difficult to see that this question is equivalent to the problem of minimizing t4 (T ) given t3 (T ), which is analogues to an interesting question about graphs: Let 2 ≤ s < r, given p(Ks , G) how small can p(Kr , G) be? (This question is stated in its 32

7

general form in [11] though it was probably posed earlier.) Razborov’s recent solution for s = 2, r = 3 [16] was a major achievement in local graph theory. The problem was subsequently solved for s = 2, r = 4 by Nikiforov [13], and for s = 2, and general r by Reiher [17]. To the best of our knowledge, the problem remains open for s ≥ 3. Proof of Proposition 3.3. Inequality (1): Recall that t3 ≥ 34 , and c3 ≤ 14 . The inequality follows, since, c3 = 2c4 +l44 +w4 . This holds with equality for the circular tournaments C, for which t4 = c4 = 12 , and l4 = w4 = 0. Inequality (2) follows by applying the inequality t3 ≥ 43 to the out-set of every vertex. To P d+ (x)
transitive 4-vertex subtournaments, count for see that there are always at least 34 3 each vertex x, the number of transitive triangles among the d+ (x) out-neighbors of x. The
inequality follows now from the convexity of the function 3t . Equality holds, e.g., for random tournaments.
Inequalities (3) and (4) are equivalent, of course. We prove the latter. Clearly n4 (t4 + w4 ) =
P d+ (x) . Again a simple convexity argument yields the inequality and equality holds for x∈V 3 random tournaments. Inequality (5): Note that two cyclic triangles sharing a common edge necessarily form a C4 . Let us denote by S the set of cyclic triangles in an n-vertex tournament T , and for an edge

P |Se |/(n)

n n
P |Se |
n n n 3c3 ( 3 )/( 2 ) 2 e, Se = {s ∈ S : e ⊂ s}. Then c4 4 = ≥ = = e 2 2 2 2 2
2 n (6 + o(1))c3 4 . The inequality follows.

4

Further directions and discussion

The following questions suggest themselves: 1. Is there some  > 0 such that every graph family with p0 , p3 < 18 +  is 4-universal? As observed by Mykhaylo Tyomkyn (personal communication), no such condition yields 5-universality, see below. 2. Is there some  > 0 such that every graph family G with p(K4 , G), p(K 4 , G) < l-universal for some values of l ≥ 3?

1 64

+  is

3. What are the triples k, l, r for which there exists an  > 0 such that the conditions k r p(Kk , G) < 2−(2) +  and p(K r , G) < 2−(2) +  imply l-universality? 4. Is there some  > 0 such that every tournaments family with t4 < 5-universal? What about l-universality for bigger l? 5. Does t5 <

5! 210

+  imply l-universality for some values of l ≥ 4?

33

8

3 8

+  is necessarily

6. For which integers k, l does there exist an  > 0 such that every tournament satisfying tk < k!k +  is l-universal? (Here tk is the proportion of transitive k-vertex subtourna2(2) ments). 7. Jacob Fox has raised the question whether problem 1.4 can have a positive answer only with l = O(k). As he pointed out, this would follow from the existence of a large k k-clique-free graph G with p(K k , G) < 2−(2) + o|G| (1). We note that it is an old and intriguing problem how small p(K k , G) can be for a large k-clique-free graph. See [4] and the recent work [14]. 8. What are the possible values of tk (T ), given the value of tl (T )? Here T is a family of tournaments and k > l ≥ 3 are integers. The first interesting (and open) case is k = 4, l = 3. 9. In this article we discuss how the paucity of small homogeneous sets implies universality in graphs and in tournaments. It is conceivable that these two problem sets can be connected, perhaps in the spirit of Alon, Pach and Solymosi [2], but we do not know how or whether this can be done. Specifically, can some connection can be established between items 3 (say, with k = r) and 6 above? We now present Jacob Fox’s proof of Proposition 1.6 which uses the following result of Pr¨omel and R¨odl. For a simpler proof of the results from [15] with improved constants see [6]. Proposition 4.1. For every a > 0 there is a b > 0 such that every n-vertex graph G with α(G), ω(G) < a log n is b log n-universal. Proof of Proposition 1.6. Let G ∈ G, be an n-vertex graph. Select H as a random subgraph of G with m = 2k/4 vertices. The expected number of k-cliques and k-anticliques in H is at
k most m (p(Kk , G) + p(K k , G)) ≤ mk · 2 · (2−(2) +  + on (1)) which can be made smaller than 1 k by making k large enough. Therefore, such an H exists with α(H), ω(H) < k. Proposition 4.1 implies that H is ck-universal, where c is a constant. This clearly implies that G is ck-universal, as claimed. Two comments are in order here: (i) The above argument applies, with some minor modifications to item 3 above and yields that if r and k are of the same order, then l could be of the same order as well. (ii) Jacob Fox has pointed out that the methods of [6] can be adapted to yield an analogue of the simple proof for Proposition 4.1 for tournaments, namely Proposition 4.2. For every a > 0 there is a b > 0 such that every n-vertex tournament T with tr(T ) < a log n is b log n-universal. From this we can easily deduce: 34

9

k

Lemma 4.3. Every family of tournaments T for which tk (T ) < k!2−(2) +  is ck-universal for some absolute c > 0. Proof. Apply the above proof of Proposition 1.6 to an arbitrary large T ∈ T . It shows that a random subset S of 2k/4 vertices in T contains no transitive k-vertex subtournament. By Proposition 4.2 the tournament induced on S, and therefore the whole of T is ck universal. Mykhaylo Tyomkyn (personal communication) found the following recursive construction which is not 5-universal even though p0 , p3 ≤ 18 . Let G1 = C5 be the pentagon graph. To construct Gn , take 5 blocks each being a copy of Gn−1 and connect every two consecutive (modulo 5) copies by a complete bipartite graph. It is easy to see that the graph Gn is self6 1 p3 (Gn−1 ) + 25 e(Gn−1 ) where e(Gn−1 ) = 12 is the edge density complementary, and p3 (Gn ) = 25 of Gn−1 . It follows that limn p3 (Gn ) = 81 = limn p0 (Gn ). On the other hand, this family of graphs is not 5-universal. We prove by induction that Gn has no induced copy of a 5-vertex path x1 , . . . , x5 . By induction not all 5 vertices are in the same block. Also, if they all reside in different blocks then x1 and x5 are adjacent which is impossible. So let xu , xv with u < v reside in the same block and xw be in a neighboring block. Then necessarily u = w − 1, v = w + 1. But at least one of xu , xv has a nighbor other than xw and this vertex cannot be fit into any of the blocks. Notes added in proof: 1. In a follow up paper, Hefetz and Tyomkin [10] settle Problem 1 in the above list and make several additional interesting contributions to this area. 2. We have recently made some progress on Problem 8 above. We intend to publish our results soon.

5

acknowledgement

We are grateful to Jacob Fox and Mykhaylo Tyomkyn for generously sharing their insights with us.

References [1] N. Alon, E. Fischer, M. Krivelevich and M. Szegedy, Efficient testing of large graphs. Combinatorica 20 (2000), 451-476. [2] N. Alon, J. Pach, and J. Solymosi, Ramsey-type theorems with forbidden subgraphs. Combinatorica 21 (2001), 155-170.

35

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[3] D. Conlon and J. Fox, Bounds for graph regularity and removal lemmas. Geom. Funct. Anal. 22 (2012), 1192-1256. [4] P. Erd˝os, On the number of complete subgraphs contained in certain graphs. Magyar Tud. Akad. Mat. Kutat´o Int. K¨ozl. 7 (1962), 459-464. [5] P. Erd˝os and A. Hajnal, Ramsey-type theorems. Discrete Applied Mathematics 25(1989), 37-52. [6] J. Fox and B. Sudakov, Induced Ramsey-type theorems. Adv. Math. 219 (2008), no. 6, 1771-1800. [7] F. Franek and V. R¨odl, 2-Colorings of complete graphs with a small number of monochromatic K4 subgraphs. Discrete Mathematics 114 (1993) 199-203. [8] G. Giraud, Sur le probleme de goodman pour les quadrangles et la majoration des nombres de ramsey. Journal of Combinatorial Theory, Series B, 27(3): 237-253, 1979. [9] A. W. Goodman, On Sets of Acquaintances and Strangers at any Party. Amer. Math. Monthly 66, 778-783, 1959. [10] D. Hefetz and M. Tyomkyn, Universality of graphs with few triangles and anti-triangles. Preprint, arXiv:1401.5735 [math.CO]. [11] H. Huang, N. Linial, H. Naves, Y. Peled and B. Sudakov, On the 3-local profiles of graphs. Preprint, arXiv:1211.3106 [math.CO]. [12] H. Huang, N. Linial, H. Naves, Y. Peled and B. Sudakov, On the densities of cliques and independent sets in graphs. Preprint, arXiv:1211.4532 [math.CO]. [13] V. Nikiforov, The number of cliques in graphs of given order and size. Trans. Amer. Math. Soc, 363(3): 1599-1618, 2011. [14] O. Pikhurko and E. Vaughan, Minimum number of k-cliques in graphs with bounded independence number. To appear in Combinatorics, Probability and Computing. [15] H. Pr¨omel and V. R¨odl, Non-Ramsey graphs are c log(n)-universal. J. Combin. Theory Ser. A 88 (1999) 379-384. [16] A. Razborov, On the Minimal Density of Triangles in Graphs. Combinatorics, Probability and Computing, Vol. 17, No 4, 2008, pages 603-618 [17] C. Reiher, The Clique Density Theorem. Preprint, arXiv:1212.2454 [math.CO]. [18] K. Sperfeld, On the minimal monochromatic K4 -density. Preprint, arXiv:1106.1030 [math.CO]. 36

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[19] A. Thomason, Graph products and monochromatic multiplicities. Combinatorica 17 (1997) 125-134.

37

12

On the number of 4-cycles in a tournament Avraham Morgenstern†

Nati Linial∗

May 22, 2014 Abstract If T is an n-vertex tournament with a given number of 3-cycles, what can be said about the number of its 4-cycles? The most interesting range of this problem is where T is assumed to have c ⋅ n3 cyclic triples for some c > 0 and we seek to minimize the number of 4-cycles. We conjecture that the (asymptotic) minimizing T is a random blow-up of a constant-sized transitive tournament. Using the method of flag algebras, we derive a lower bound that almost matches the conjectured value. We are able to answer the easier problem of maximizing the number of 4-cycles. These questions can be equivalently stated in terms of transitive subtournaments. Namely, given the number of transitive triples in T , how many transitive quadruples can it have? As far as we know, this is the first study of inducibility in tournaments.

1

Introduction and notation

1.1

Notation

For tournaments T, H, let pr(H, T ) be the probability that a random set of ∣H∣ vertices in T spans a subtournament isomorphic to H. For an infinite family of tournaments T , let pr(H, T ) = limT ∈T ,∣T ∣→∞ pr(H, T ), assuming the limit exists. (Nonexistence of the limit may be repaired, of course, by passing to an appropriate subfamily). We denote the transitive m-vertex tournament by Tm , and the 3-vertex cycle by C3 . There are four isomorphism types of 4-vertex tournaments, see Figure 1. • C4 which is characterized by having a directed 4-cycle, • The transitive T4 , • W , a cyclic triangle and a sink, • L, a cyclic triangle and a source.

38

1

Figure 1: T4 , C4 , W, L Denote r(T ) = pr(R, T ) for any letter r ∈ {c3 , c4 , tk , w, l} (e.g., c3 (T ) is the limit proportion of cyclic triangles in members of T ). We omit T when appropriate. We will always restrict ourselves to families for which all the relevant limits exist, though we do not bother to mention this any further. In [7] we initiated the study of 4-local profiles of tournaments, namely the set P = {(t4 (T ), c4 (T ), w(T ), l(T ))∣T is a family of tournaments for which all the limits exist} ⊆ R4 . Here we continue with these investigations.

1.2

Our questions

In studying the set P of 4-local profiles of tournaments, it is of interest to understand its projection to the first two coordinates, which raises Problems 3 and 6 below. We are, in fact, interested in all the following six problems, but as we show below, they are interdependent. 1 Maximize c4 (T ) when c3 (T ) is set. 2 Maximize t4 (T ) when t3 (T ) is set. 3 Maximize c4 (T ) when t4 (T ) is set. 4 Minimize c4 (T ) when c3 (T ) is set. 5 Minimize t4 (T ) when t3 (T ) is set. 6 Minimize c4 (T ) when t4 (T ) is set. School of Computer Science and engineering, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Email: [email protected]. Research supported in part by an ERC grant 339096. † Einstein Institute of mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Email: [email protected] ∗

39

2

Proposition 1.1. Problems 1 to 3 are equivalent in the sense that the solution of any one of them can be transformed into a solution for the other two. Likewise Problems 4 to 6 are equivalent. To prove this proposition we need Observation 1.2. t4 − c4 = 1 − 4c3 Proof. We count cyclic triangles in 4-vertex tournaments. There are none in T4 , two in C4 and one each in W and L. Therefore the number of cyclic triangles in an n-vertex tournament (n4 ). The claim follows, since t4 + c4 + w + l = 1. satisfies c3 (n3 ) = 2c4 +w+l n We can now prove Proposition 1.1. Proof of Proposition 1.1. Obviously, t3 + c3 = 1. Combined with Observation 1.2 this already proves that Problems 1 and 2 and Problems 4 and 5 are equivalent. To see that e.g. Problems 5 and 6 are equivalent, note that Problem 5 is equivalent to maximizing t3 for given t4 , or, equivalently, to minimizing c3 given t4 . The equivalence follows by Observation 1.2. Problems 1 to 3 are rather straightforward and we proceed to solve them. Problems 4 to 6 are deeper. By the equivalence proved above, the discussion is restricted to problem 4. We state a conjecture on the solution of this problem and prove a lower bound. This problem raises interesting structural limitations on tournaments, on which we elaborate in Section 2. We defer the technical proofs to Section 3 and in Section 4 we offer some further directions. The three regions {(t3 , t4 )}, {(c3 , c4 )}, {(t4 , c4 )} of the realizable pairs of parameters are illustrated in Figures 2 to 4. We finally note that the planar sets {(t3 (T ), t4 (T ))}, {(c3 (T ), c4 (T ))}, {(t4 (T ), c4 (T ))} are simply connected. This shows that these sets coincide with the bounded regions in Figures 2 to 4. Lemma 1.3. The set of all pairs (c3 (T ), c4 (T )) is simply connected. Here T is an arbitrary family of tournaments for which these limits exist. The arguments introduced in Proposition 1.1 yield the same conclusion for the sets {(t3 , t4 )}, {(t4 , c4 )}. We prove the lemma in Section 3.

2 2.1

Results and conjectures The maximum

In this subsection we solve Problems 1 to 3: Observation 2.1. The following inequalities hold in all tournaments. These inequalities are tight. 40

3

1

0.9

0.8

t4

0.7

0.6

0.5

0.4

0.75

0.8

0.85

0.9

0.95

1

t

3

Figure 2: The boundary of the set {(t3 (T ), t4 (T ))}. The lower curve is conjectured. • c4 ≤ 2c3 . • t4 ≤ 2t3 − 1. • c4 ≤ min{t4 , 1 − t4 }. Proof. Clearly t4 + c4 ≤ 1, since t4 + c4 + w + l = 1. Also, as we saw t4 − c4 = 1 − 4c3 and c3 + t3 = 1. In addition it is well-known and easy to show that c3 ≤ 41 . All the inequalities follow. To show that the first two inequalities are tight, we construct (Section 3) tournaments with w = l = 0 for all values of c3 ≤ 41 . This shows as well the tightness of the third inequality when t4 ≥ 12 . For t4 ≤ 12 we need a different construction which satisfies c4 = t4 , which we also do in Section 3.

2.2

The minimum

Problems 4 to 6 are more involved. We focus on Problem 4. To derive an upper bound for Problem 4 we introduce the random blow-up of a k-vertex tournament H. Associated with H and a probability vector (w1 , . . . , wm ) is an infinite family of tournaments T = T (H; w1 , . . . , wm ) whose n-th member has vertex set ⊍{Vi ∣i ∈ H} where ∣Vi ∣ = ⌊wi n⌋. If (i → j) ∈ E(H) then 41

4

0.5 0.45 0.4 0.35 0.3

c4

0.25 0.2 0.15 0.1 0.05 0 0

0.05

0.1

0.15

0.2

0.25

c

3

Figure 3: The boundary of the set {(c3 (T ), c4 (T ))}. The lower curve is conjectured. there is an edge (u → v) from every u ∈ Vi to every v ∈ Vj . The subtournament on each Vi is random. In the balanced case w1 = w2 = . . . = wk = m1 , we use the shorthand T (H). We can now state our conjecture. Conjecture 2.2. The minimum of c4 , given c3 is attained by a random blow-up of a transitive tournament Tm . Lemma 2.5 below says that among all such tournaments of given c3 , the smallest c4 is attained by taking m as small as possible and w1 = w2 = . . . = wm−1 ≥ wm . When c3 = 4r12 the random blow-up that minimizes c4 is the balanced blow-up of Tr . It is conceivable that this case of the conjecture should be easier to handle. When c3 = 14 and k = 1 this reduces to the well-known fact that t4 is minimized by a random tournament. (Recall that given c3 , minimization of c4 and of t4 are equivalent). In this article we settle as well the 1 case c3 = 16 and m = 2. Note that, if the conjecture is indeed true, then there is no simple expression for min c4 in terms of c3 . We reproduce a proof from [7], that we later (Lemma 2.4) improve. Proposition 2.3. c4 ≥ 6c23 . 42

5

0.5 0.45 0.4 0.35 0.3

c4

0.25 0.2 0.15 0.1 0.05 0 0.4

0.5

0.6

0.7

0.8

0.9

1

t

4

Figure 4: The boundary of the set {(t4 (T ), c4 (T ))}. The lower curve is conjectured. Proof. For an edge e = uv in a tournament T , let xe be the probability that the triangle uvw is cyclic when the vertex w is selected uniformly at random. We define the random variable X on E(T ) with uniform distribution that takes the value xe at e ∈ E(T ). Clearly EX = c3 + o∣T ∣ (1). But a 4-vertex tournament is isomorphic to C4 iff it contains two cyclic triangles with a common edge. Consequently, E(X 2 ) = c64 + o∣T ∣ (1). The proposition simply says that V ar(X) ≥ 0. Consequently, our main problem is to find the smallest possible variance V ar(X) for given E(X). Conjecture 2.2 and Lemma 2.4 below are some quantitative forms of the assertion that when 0 < c3 < 14 , cyclic triangles cannot be uniformly distributed among the edges. We presently have no conceptual proof of this claim, and we must resort to flag algebra methods, which unfortunately offer no intuition as to the reason that this statement is true. Here is another curious aspect of this problem. Define ϕT (x) ∶= Pr(X ≥ x) and let f ∶= lim sup∣T ∣→∞ ϕT . For all we know, f may be discontinuous. To see this note that f ( 13 ) ≥ 23 where 32 is the value that is attained by balanced blow-ups of C3 . We suspect that f ( 31 +) is strictly smaller than 23 . In fact, the best lower bound that we have is f ( 13 +) ≥ 94 which is attained by an imbalanced blow-up of C3 . We turn next to apply Razborov’s flag-algebra method [6] which yields a lower bound that 1 is not far from the conjectured value. In particular, it proves Conjecture 2.2 for c3 = 16 . 43

6

0.06 Our best construction (optimal?) numerical FA lower bound Lower bound (Lemma 2.4)

0.05

0.04

c4

0.03

0.02

0.01

0

0

0.01

0.02

0.03

c

0.04

0.05

0.06

0.07

3

Figure 5: Lower bound from numerical application of flag algebras, compared with proven lower bound and our best construction, which is conjectured to be optimal. To improve visibility we present only the range c3 ∈ [0, 0.07]. Numerical lower bound and the construction 1 seem to coincide for c3 ≥ 16 . Lemma 2.4. c4 ≥

18c23 1+8c3 .

See Figure 5 for a comparison between this bound and Conjecture 2.2. Using available computer software, we were able to get further numerical evidence which 1 1 , 4 , and the true minimum of c4 is closer to indicates that Lemma 2.4 is not tight for c3 ≠ 0, 16 the conjectured value. The results are graphically presented in Figure 5 and the method of computation is explained in Appendix A. Concluding this section, we formulate the following analytic lemma. It states that among all blow-ups considered in Conjecture 2.2 the best one is a blow-up of a transitive tournament of least possible order, with equal vertex weights, except possibly one smaller weight. Lemma 2.5. Fix any 0 < C < 1 and consider all probability vectors w satisfying ∑ wi3 = C. The minimum of ∑ wi4 among such vectors is attained by letting w1 = . . . = wm−1 ≥ wm > 0 with the smallest possible m. The relevance of the lemma in the setting of Conjecture 2.2, is that c3 (T ) = 41 ∑ wi3 , and c4 (T ) = 38 ∑ wi4 where T = T (Tm ; w1 , . . . , wk ). 44

7

3

Proofs

Proof of Lemma 1.3. We will show that the set {(c3 , c4 )} is vertically convex. Let T1 , T2 be two families with c3 (T1 ) = c3 (T2 ) and c4 (T1 ) < c < c4 (T2 ). We construct an n-vertex tournament T with c3 (T ) = c3 (T1 ) + on (1) and c4 (T ) = c + on (1). Let 0 ≤ p, α ≤ 1 be two constant parameters. Choose T1 ∈ T1 on αn vertices (we can choose a random subtournament of a larger member if T1 has no member of this order). Let T2 ∈ T2 of order (1 − α)n. Let T = T1 ⊍ T2 , where for x ∈ T1 and y ∈ T2 there is an edge x → y with probability p and y → x with probability 1 − p. We compute c3 (T ) = α3 c3 (T1 ) + (1 − α)3 c3 (T2 ) + 3α(1 − α)p(1 − p) + o(1). Choose p such that p(1 − p) = c3 (T1 ) = c3 (T2 ) and then c3 (T ) = c3 (T1 ) + o(1). In computing c4 (T ), several terms come in, each up to +o(1) • α4 c4 (T1 ) for quadruples contained in T1 • (1 − α)4 c4 (T2 ) all in T2 • 6α2 (1 − α)2 (p(1 − p) + 2p2 (1 − p)2 ) two in each • 4α3 (1 − α)(c3 (T1 )3p(1 − p) + (1 − c3 (T1 ))p(1 − p)) three in T1 and one in T2 • 4α(1 − α)3 (c3 (T2 )3p(1 − p) + (1 − c3 (T2 ))p(1 − p)) one and three. Consequently c4 (T ) is expressed (up to an additive o(1) term) as a degree four polynomial in α which for 1 ≥ α ≥ 0 takes every value between c4 (T1 ) and c4 (T2 ). Completing the proof of tightness in Observation 2.1. Let us recall the well-known cyclic tournaments (see e.g., [7]). Place an odd number of vertices equally spaced along a circle, and x → y is an edge if the shorter arc from x to y is clock-wise. We are now ready to construct tournaments with the desired parameters. • Tournaments with arbitrary 0 ≤ c3 ≤ 41 , and w = l = 0: Fix some n2 ≤ s ≤ n. Let T be the tournament with vertex set 1, 2, . . . , n, where x → y for 1 ≤ x < y ≤ n, iff y ≤ x + s. We claim that w(T ) = l(T ) = 0. For suppose that x → y → z → x is a cyclic triangle in T and there is some vertex w with either w → x, y, z or w ← x, y, z. w.l.o.g. x < y, z and it follows that x < y ≤ x + s < z ≤ y + s. If w < x, then w → x since s ≥ n2 , but z → w. Likewise we rule out the possibility that w > z, i.e., necessarily x < w < z. If x < w < y then necessarily x → w → y. Likewise, y < w < z implies y → w → z. For n → ∞ odd and s = n2 this yields the cyclic tournaments and c3 = 14 . when s = n we obtain transitive tournaments. As s varies we cover the whole range 0 ≤ c3 ≤ 41 . • For t ∈ [ 38 , 21 ], we construct a family T with t4 (T ) = c4 (T ) = t: Fix some 0 ≤ p ≤ 21 . We construct T from the cyclic tournaments by flipping each edge 45

8

independently with probability p. As we show below, c3 (T ) = 14 , so by Observation 1.2, t4 (T ) = c4 (T ). When p = 0 we have the cyclic tournament with t4 = c4 = 12 and when p = 21 we have a random tournament with t4 = c4 = 83 . The claim follows by continuity. To see that c3 (T ) = 41 , note that almost surely all vertex outdegrees in T ∈ T equal n 2 ± o(n). The claim follow by a standard Goodman-type argument. Proof of Lemma 2.4. We define the random variables X and Y over E(T ) with uniform distribution. For e = {v1 → v2 } ∈ E(T ) we define: • X(e) is the probability that {v1 , v2 , v3 } is a cyclic triangle in T , where the vertex v3 is chosen uniformly at random. • Y (e) the probability that {v1 → v3 } ∈ E(T ) and {v3 → v2 } ∈ E(T ), where the vertex v3 is chosen uniformly at random. It is not hard to verify the following expectations: E(X) = c3 , E(Y ) = t33 . E(X 2 ) = c64 , +c4 E(Y 2 ) = t64 and E(X ⋅ Y ) = t412 . 3 We define Z = 1 + 2(X − Y ) and conclude that E(Z 2 ) = 1+8c and E(X ⋅ Z) = c3 . By 3 1+8c3 2 2 2 2 Cauchy-Schwarz c3 = E (X ⋅ Z) ≤ E(X )E(Z ) = c4 18 . Remark 3.1. Proper disclosure: The above derivation could not have been carried out without seeing what flag-algebra calculations yield. m 3 m 4 Proof of Lemma 2.5. We wish to minimize ∑m 1 wi under the constraints ∑1 wi = 1 and ∑1 wi = C (for given m). We assume that all wi are positive, since zero wi ’s can be removed with smaller m. A Lagrange multipliers calculation yields that wi3 = λwi2 + µ for all i and for some constants λ and µ. The cubic polynomial x3 − λx2 − µ has at most two positive roots since the linear term in x vanishes. Therefore the coordinates of the optimal w must take at most two distinct values. Assume towards contradiction that x > y > 0 appear as coordinates in w with y repeated at least twice. We will replace three of w’s coordinates (x, y, y) while preserving ∑m 1 wi and m 4 m 3 w , and reducing w = C. ∑1 i ∑1 i √ We replace (x, y, y) by either (s, t, 0) or (s, s, t) where s ≥ t ≥ 0. First, if y ≤ 5−1 ⋅ x, we 4 prove the existence of s ≥ t ≥ 0 s.t.

• x + 2y = s + t. • x3 + 2y 3 = s3 + t3 . • x4 + 2y 4 > s4 + t4 . Substitute t = x + 2y − s in the second equation: x3 + 2y 3 = s3 + (x + 2y − s)3 , which can be rewritten as (x + 2y)s2 − (x + 2y)2 s + 2y(x + y)2 = 0. This quadratic has real roots iff 46

9

√ D = (x + 2y)4 − 8y(x + y)2 (x + 2y) ≥ 0 which holds iff x ≥ (1 + 5)y, the range that we consider. Moreover, when D ≥ 0, both roots are positive, since the quadratic has a positive constant term and a negative linear term. This proves the existence of s ≥ t ≥ 0 satisfying the first two conditions. 8 +6(x+2y)4 D+D 2 The sum of the fourth powers of the roots of this quadratic is s4 + t4 = (x+2y) 8(x+2y) . 4 4 4 2 6 3 2 2 4 Thus, it suffices to show that 8(x +2y )(x+2y) > 8(x+2y) −64y(x+2y) (x+y) +64y (x+y) which is easily verified by expanding all √terms. In the complementary range x > y ≥ 5−1 4 ⋅ x we find s ≥ t ≥ 0 s.t. • x + 2y = 2s + t. • x3 + 2y 3 = 2s3 + t3 . • x4 + 2y 4 > 2s4 + t4 . We substitute t = x + 2y − 2s in the second equation: x3 + 2y 3 = 2s3 + (x + 2y − 2s)3 , or, equivalently, 0 = s3 − 2(x + 2y)s2 + (x + 2y)2 s√− y(x + y)2 = (s − y)(s2 − 2xs − 3ys + (x + y)2 ). Thus, s2 − (2x + 3y)s + (x + y)2 = 0, and s = Clearly, s ≥ t. It remains to compute

2x+3y− y(4x+5y) 2

> 0. Now t = x + 2y − 2s ≥ 0 iff y ≥

√ 5−1 4

⋅ x.

√ x4 +2y 4 −2s4 −t4 = (8x3 +50x2 y+86xy 2 +45y 3 ) y(4x + 5y)−(2x4 +52x3 y+180x2 y 2 +232xy 3 +101y 4 ) and show that this is positive. To this end we must prove that y(4x + 5y)(8x3 + 50x2 y + 86xy 2 + 45y 3 )2 − (2x4 + 52x3 y + 180x2 y 2 + 232xy 3 + 101y 4 )2 > 0. This expression can be written as 4(x − y)3 (19y 5 + 73xy 4 + 98x2 y 3 + 54x3 y 2 + 9x4 y − x5 ) which is positive, since 9x4 y − x5 > 0.

4

Further Directions • Many basic open questions on the local profiles of combinatorial objects are still open. Thus, it is still unknown whether the set of k-profiles of graphs is a simply connected set. Similar issues were already raised in the pioneering work of Erd˝os, Lov´asz and Spencer [3], and remains open. The analogous question for tournaments is open as well. We do not know if the set of k-local profiles of tournaments is convex. We don’t know it even for k = 4, and we are not sure what the right guess is. As first observed in [3] the analogous question is answered negatively for graphs. On the other hand, for trees the answer is positive [1]. 47

10

• We recall the random variable X - the fraction of cyclic triangles containing a randomly chosen edge. It would be desirable to give a direct proof that Var(X) > 0 for all 0 < c3 < 41 . • In Section 2 we defined the function f (x) = lim sup∣T ∣→∞ pr(X ≥ x). What can be said about f ? In particular, is it continuous? Is it continuous at 31 ? • The conjectured extreme construction for Problem 4 is particularly simple when c3 = 4k12 for integer k. We were able to settle this case for k = 1, 2. Thus, the first open case is 1 c3 = 36 . • To what extent can the lower bound in Lemma 2.4 be improved using higher order flags? 1 In particular, Figure 5 suggests that our construction is optimal for c3 ≥ 16 . Can the optimum for this range be established using flags of order 6? • Here we have studied the set {(t3 (T ), t4 (T ))}. We would like to understand the relationships among higher tk ’s as well. • Obviously, we would be interested in further describing the set of 4-profiles of tournaments. • The powerful method of flag algebras remains mysterious, and it would be desirable to have more transparent local methods. Lemma 2.4 and the stronger Conjecture 2.2 offer concrete challenges for such methods. • Associated with every tournament T is a 3-uniform hypergraph whose faces are the cyclic triangles of T . This hypergraph clearly does not contain a 4-vertex clique and this was used in [4] to deduce a lower bound on some hypergraph Ramsey numbers. We wonder about additional structural properties of such 3-uniform hypergraphs. Specifically, – Can such hypergraphs be recognized in polynomial time? • Lemma 2.5 is the case p = 3, q = 4 of the following natural sounding question. Find the smallest q-norm among all probability vectors w of given p-norm, where q > p ≥ 2 are integers. Is it true that all optimal vectors have the form w1 = . . . = wm−1 ≥ wm , with the least possible m? Clearly our method of proof is too ad-hoc to apply in general.

5

acknowledgement

We are grateful to Yuval Peled for many useful discussions about flag algebras and for his help with the computer calculations described in Appendix A. Thanks are also due to Gideon Schechtman for discussions concerning the proof of Lemma 2.5.

48

11

References [1] S. Bubeck, N. Linial, On the local profiles of trees. Preprint, arXiv:1310.5396 [math.CO]. [2] CVX Research, Inc., CVX: Matlab software for disciplined convex programming, version 2.0. http://cvxr.com/cvx, April 2011. [3] P. Erd˝os, L. Lov´asz, J. Spencer, Strong independence of Graphcopy functions. Graph Theory and Related Topics (Academic Press, New York, 1979), 165-172. [4] P. Erd˝os, A. Hajnal, On Ramsey like theorems, problems and results. Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972). 1972. [5] H. Huang, N. Linial, H. Naves, Y. Peled, B. Sudakov, On the 3-local profiles of graphs. Journal of Graph Theory, Volume 76, 236–248, 2014. [6] A. Razborov, Flag Algebras. Journal of Symbolic Logic, Vol. 72, No 4, 2007, pages 12391282. [7] N. Linial, A. Morgenstern, Graphs with few 3-cliques and 3-anticliques are 3-universal. Journal of Graph Theory, to appear, arXiv:1306.2020.

A

Computer generated lower bounds

We have used the flag algebra method as explained in Section 4 of [5]. Using flags of size 3 over the (only) type of order 2 yields Lemma 2.4. Using flags of size 4 over the same type we get a 16 × 16 PSD matrix whose entries are bilinear expressions in the coordinates of a large tournament’s 6-profile. We used the cvx SDP-solver [2] to obtain the results presented in Figure 5. Working with larger flags may clearly yield better estimates but limited computational resources have stopped from reaching beyond size 4-flags.

49

12

‫בכל הקואורדינטות‪ (.‬אנו מנסחים "גרסה מקומית" של ההשערה‪ :‬מחליפים את החסם על‬ ‫גודל התת קבוצה ההומוגנית הגדולה ביותר בחסם על מספר התת קבוצות ההומוגניות מגודל‬ ‫קבוע‪ .‬אנו מוכיחים שאם שכיחותם של תתי גרפים שלמים מגודל ‪ 3‬וגם השכיחות של תתי‬ ‫גרפים ריקים מגודל ‪ 3‬שניהם קטנים מ ‪ 0.159...‬אז הגרף הוא ‪3‬־אוניברסלי )בתנאי שמספר‬ ‫הקדקודים בגרף גדול מספיק(‪ .‬עבור תחרויות אנו מוכיחים שאם שיעורן של התת תחרויות‬ ‫הטרנזיטיביות מגודל ‪ 4‬קטן מ  ‪ 21 −‬אז התחרות היא ‪4‬־אוניברסלית‪.‬‬ ‫בפרק השלישי אנו ממשיכים בחקר קבוצת הווקטורים ב־ ‪ R4‬המתקבלים כ ‪4‬־פרופילים‬ ‫של תחרויות גדולות‪ .‬אנו מתרכזים בהטלה על שתי קואורדינטות‪ :‬ההסתברות להמצאותה‬ ‫של ‪ T4‬־ תת תחרות טרנזיבית מגודל ‪ ,4‬וזו של ‪ C4‬התת תחרות ההמילטונית )היחידה(‬ ‫מגודל ‪ .4‬אנו מעלים כמה שאלות השקולות לשאלות על ההטלה האמורה‪ .1 :‬בהינתן‬ ‫מספר המשולשים הציקליים‪ ,‬איך ממזערים את מספר המופעים של ‪ .2 ?C4‬בהינתן מספר‬ ‫המשולשים הטרנזיטיביים‪ ,‬איך ממזערים את מספר המופעים של ‪ .3 ?T4‬נתייחס לקבוצת‬ ‫הצלעות של תחרות נתונה ‪ T‬כמרחב הסתברות בעל התפלגות אחידה‪ .‬נתבונן במשתנה‬ ‫המקרי ‪ X‬המתאים לצלע ‪ e‬את ההסתברות שקדקוד מקרי ‪ v‬יוצר יחד עם ‪ e‬משולש ציקלי‬ ‫ב־ ‪ .T‬בהנתן )‪) E(X‬מה ששקול לקביעת שיעורם של המשולשים הציקליים(‪ ,‬איך ניתן למזער‬ ‫את השונות )‪?V ar(X‬‬ ‫אנו מוכיחים שהמשולשים הציקליים אינם יכולים להמצא בפיזור אחיד על פני הצלעות‬ ‫אלא אם התחרות היא )כמעט( טרנזיטיבית או "פסאודו־מקרית" )במובן זה ששיעור המשולשים‬ ‫הציקליים הוא )‪.( 14 − o(1‬‬ ‫אנו מנסחים השערות לגבי כל השאלות לעיל‪ ,‬ומוכיחים חסמים שכמעט תואמים לערך‬ ‫המשוער‪ .‬החסמים מופקים בשיטה של אלגבראות הדגלים‪ ,‬אך ניתן להוכיחם בשיטות‬ ‫אלמנטריות‪ .‬אנו מציגים מספר שאלות לגבי המבנה של תחרויות גדולות‪.‬‬

‫‪References‬‬ ‫‪[1] J. Moon, Topics on tournaments. Holt, Rinehart and Winston, New‬‬ ‫‪York-Montreal, Que.-London 1968 viii+104 pp.‬‬ ‫‪[2] A. Thomason, A disproof of a conjecture of Erd®s in Ramsey theory,‬‬ ‫‪J. London Math. Soc. (2), 39(2) (1989), 246255.‬‬ ‫‪[3] A. Thomason. Graph products and monochromatic multiplicities.‬‬ ‫‪Combinatorica, 17(1):125134, 1997.‬‬ ‫‪[4] N. Alon, J. Pach, and J. Solymosi, Ramsey-type theorems with for‬‬‫‪bidden subgraphs. Combinatorica 21 (2001), 155-170.‬‬

‫‪2‬‬

‫‪50‬‬ ‫‪4‬‬

‫תקציר‬

‫תחרות היא אוריינטציה של גרף שלם‪ .‬המחקר בתחרויות הוא ענף מרכזי של תורת‬ ‫הגרפים‪ ,‬וכבר בשנות ה־‪ 60‬נכתבה מונוגרפיה ]‪ [1‬על הנושא‪ .‬מעבר לעניין בתחרויות‬ ‫כאובייקט מתמטי מעניין ושימושי בפני עצמו‪ ,‬יש להן קשר מסקרן ומסתורי עם בעיות‬ ‫יסודיות בתורת הגרפים‪ .‬יש דוגמאות רבות לקשר זה‪.‬‬ ‫במקרים מסוימים בעיה נעשית קלה כששואלים אותה על תחרויות‪ .‬לדוגמה‪ ,‬איך למזער‬ ‫את מספר תתי המבנים ההומוגניים מגודל קבוע ‪ ?k‬בגרף מדובר על קליקות ואנטי‬ ‫קליקות ובתחרות ־ על תת־תחרות טרנזיטיבית‪ .‬קל להוכיח שתחרויות מקריות משיגות‬ ‫)אסימפטוטית( את המינימום האפשרי‪ .‬כש ‪ k = 3‬הראה גודמן תופעה דומה בגרפים‪ ,‬מה‬ ‫שהביא את ארדש לשער זאת לגרפים ול־‪ k‬כללי‪ ,‬אך ההשערה הזו הופרכה לכל ‪k ≥ 4‬‬ ‫]‪.[3 ,2‬‬ ‫לפעמים הבעיה בתחרויות ובגרפים שקולות זו לזו‪ ,‬לדוגמה‪ ,‬השערת ארדש־היינל ]‪.[4‬‬ ‫לפעמים אנו מרגישים שקיים קשר עמוק בין הבעיות האנלוגיות בתחרויות ובגרפים‪ ,‬אך איננו‬ ‫מסוגלים להצביע על קשר כזה‪ ,‬לדוגמה‪ ,‬חסמים על מספרי רמזי; ידוע שכל גרף מסדר ‪ n‬מכיל‬ ‫קליקה או אנטי קליקה מגודל ‪ 12 log n‬ומצד שני בגרפים מקריים אין קליקה או אנטי קליקה‬ ‫גדולה מ־‪ .2 log n‬משפט ארדש־מוזר‪ :‬קל לראות שבכל תחרות מסדר ‪ n‬יש תת־תחרות‬ ‫טרנזיטיבית על ‪ log n‬קדקודים‪ ,‬אבל בתחרות מקרית אין תת־תחרות טרנזיטיבית גדולה‬ ‫מ־‪ .2 log n‬למרות הדמיון בין התופעות לא ידוע קשר פורמלי ביניהן‪.‬‬ ‫דוגמה מעניינת שאנו מעלים בעבודה זו היא שאלת הקמירות של הפרופילים המקומיים‪.‬‬ ‫ידוע כבר משנות ה־‪ 70‬שאוסף הפרופילים המקומיים של גרפים אינו קבוצה קמורה‪ ,‬אך לא‬ ‫ברורה התשובה לפרופילים המקומיים של תחרויות‪.‬‬ ‫הפרק הראשון בתזה מהווה תרומה לחקר תחרויות רב־מימדיות‪ .‬למרות שגרפים רב־‬ ‫מימדיים הם נושא מרכזי למחקר כבר שנים רבות‪ ,‬רק מעט מאוד ידוע על תחרויות רב־‬ ‫מימדיות‪ .‬אנו בוחנים כמה הכללות אפשריות למושג של תחרות אציקלית‪ ,‬ומבררים את‬ ‫היחסים ביניהן‪ .‬אנו מוכיחים אנלוגים רב־מימדיים לכמה משפטים ידועים‪ .1 :‬מספר‬ ‫‪n‬‬ ‫התחרויות האציקליות ה־‪d‬־מימדיות על ‪ n‬קדקודים הוא ) ‪ .2 .(Θ(n))( d‬משפט ארדש־מוזר‪.‬‬ ‫אנו משתמשים בתוצאות שהוכחנו כדי להכליל טיעון של ארדש והיינל בתורת רמזי‪.‬‬ ‫בפרק השני אנו חוקרים את הפרופילים המקומיים של גרפים ותחרויות‪ .‬ה־ ‪k‬־פרופיל‬ ‫של גרף ‪ G‬הוא וקטור שבו כל קואורדינטה מתאימה לטיפוס איזומורפיזם ‪ H‬של גרפים על‬ ‫‪ k‬קדקודים‪ ,‬ומכילה את ההסתברות שתת גרף מקרי של ‪ k‬קדקודים ב־‪ G‬איזומורפי ל־‪.H‬‬ ‫תורת הגרפים המקומית דנה בשאלות הבאות‪ .1 :‬אלו וקטורי הסתברות יכולים להתקבל‬ ‫כפרופילים מקומיים של גרפים גדולים? ‪ .2‬מה אפשר ללמוד על גרף גדול מתוך הסתכלות‬ ‫בפרופיל המקומי שלו?‬ ‫כפי שנסביר בהמשך‪ ,‬אנו מוכיחים שתנאים מסוימים על הפרופיל המקומי גוררים )גרסה‬ ‫מוחלשת של( פסאודו־מקריות‪ .‬נזכיר את השערת ארדש־היינל הטוענת שכל גרף גדול מסדר‬ ‫‪ n‬ללא תת־גרף הומוגני מגודל )‪ nΩk (1‬הוא ‪k‬־אוניברסלי )גרף הוא ‪k‬־אוניברסלי אם כל גרף‬ ‫על ‪ k‬קדקודים איזומורפי לאיזשהו תת־גרף שלו‪ .‬במילים אחרות‪ ,‬ה־‪k‬־פרופיל שלו חיובי‬

‫‪1‬‬

‫‪51‬‬ ‫‪3‬‬

1

‫עבודה זו נעשתה בהדרכתו של‬ ‫פרופסור נתי ליניאל‪.‬‬

‫‪1‬‬

‫‪52‬‬ ‫‪2‬‬

1

‫תורה מקומית ותורה רב־מימדית של תחרויות‬

‫חיבור לשם קבלת תואר דוקטור לפילוסופיה‬ ‫מאת‬ ‫אברהם מורגנשטרן‬

‫הוגש לסנט האוניברסיטה העברית בירושלים‬ ‫יולי ‪2014‬‬

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