1

2 3

THE NUMBER OF B3 -SETS OF A GIVEN CARDINALITY ˇ ¨ DOMINGOS DELLAMONICA JR., YOSHIHARU KOHAYAKAWA, SANG JUNE LEE, VOJTECH RODL, AND WOJCIECH SAMOTIJ Abstract. A set S of integers is a B3 -set if all the sums of the form a1 + a2 + a3 , with a1 , a2 and a3 2 S and a1  a2  a3 , are distinct. We obtain asymptotic bounds for the number of B3 -sets of a given cardinality contained in the interval [n] = {1, . . . , n}. We use these results to estimate the maximum size of a B3 -set contained in a typical (random) subset of [n] of a given cardinality. These results confirm conjectures recently put forward by the authors [On the number of Bh -sets, submitted].

4

1. Introduction

5

Let h 2 be an integer. A set S of integers is a Bh -set if for any z there is at most one sequence a1  · · ·  ah satisfying z = a1 + · · · + ah if we require that ai 2 S for every i = 1, . . . , h. The study of Bh -sets goes back to Sidon [14], who asked how large B2 -sets, or Sidon sets, can be if one imposes that they should be subsets of [n] = {1, . . . , n}. Let

6 7 8

Fh (n) = max{|S| : S ⇢ [n] is a Bh -set}. 9 10 11 12

In the case addressed by Sidon, that is, for h = 2, results of Chowla, Erd˝os, Singer, and Tur´ an [3, p 5, 6, 15] from the 1940s tell us that F2 (n) = (1 + o(1)) n. The case of general h is less well understood. Bose and Chowla [1] showed that Fh (n) (1 + o(1))n1/h for h 3, while an easy argument gives that, for every h 3 and large n, Fh (n)  (h · h! · n)1/h  h2 n1/h .

13 14 15

16 17

(1)

(2)

Note that, for h = 3, the first inequality in (2) gives that F3 (n)  3n1/3 for all large enough n. For general h, successively better bounds of the form Fh (n)  ch n1/h have been obtained. The latest bounds are due to Green [7], who proved that ✓ ✓ ◆ ◆ 1 3 c3 < 1.519, c4 < 1.627 and ch  h+ + o(1) log h , (3) 2e 2 where o(1) ! 0 as h ! 1. For a wealth of material on Sidon sets and on Bh -sets, the reader is referred to the classical monograph of Halberstam and Roth [8] and to a survey by O’Bryant [12]. Date: 2014/02/07, 2:59am. The second author was partially supported by FAPESP (2013/03447-6, 2013/07699-0), CNPq (308509/2007-2 and 477203/2012-4), NSF (DMS 1102086) and NUMEC/USP (Project MaCLinC/USP). The third author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future Planning (2013R1A1A1059913) and supported by the National Research Foundation of Korea(NRF) Grant funded by the Korea Government(MSIP) (No. 2013042157). The fourth author was supported by the NSF grants DMS 0800070, 1301698, and 1102086. The fifth author was partially supported by ERC Advanced Grant DMMCA and a Trinity College JRF. 1

18 19 20

A related problem to bounding Fh (n) is the problem of estimating how many Bh -sets [n] contains. In fact, this problem was raised by Cameron and Erd˝os [2] in 1990 for h = 2. Let us introduce the following definition.

21

Definition 1.1 (Zh (n), Zh (n, s)). For non-negative integers 1  s  n, let

22

Zh (n, s) = S ⇢ [n] : |S| = s and S is a Bh -set . P Furthermore, let Zh (n) = s Zh (n, s).

23 24

(4)

In view of the fact that Fh (n) = ⇥(n1/h ), one sees that c0h n1/h  log Zh (n)  Ch n1/h log n for some positive constants c0h and Ch . One now knows that, in fact, log Zh (n)  Ch0 n1/h

(5)

32

for some constant Ch0 . The case h = 2 of (5) is proved in [11] (see also [13]), and the arbitrary h case is dealt with in [4]. As it turns out, to establish (5), we considered the more refined question of estimating Zh (n, s). Roughly speaking, we obtained good bounds for Zh (n, s) for s n1/(h+1) (log n)2 and derived (5) summing over all relevant s (see [4] for details). The problem of estimating Zh (n, s) for the whole range of s is interesting in its own right, and has an application to a certain problem in probabilistic combinatorics (we shall come back to this application in Sections 2 and 7). To develop a feel for the problem of estimating Zh (n, s), let us state lower bounds for this quantity, proved in [4].

33

Proposition 1.2 (Lower bounds for Zh (n, s)). The following bounds hold for every h

25 26 27 28 29 30 31

34

35 36

(i ) There is a constant c0h > 0 such that, for all n and s, we have ✓ 0 ◆s ⌫ ch n Zh (n, s) . sh (ii ) For any we have

> 0, there is an " > 0 such that, for any s  "n1/(2h Zh (n, s)

37 38 39 40 41 42 43 44 45 46 47

2.

)s

(1

(6) 1)

and any large enough n,

✓ ◆ n . s

(7)

The lower bound in (6) may be proved coupling Bose and Chowla’s construction [1] and a simple product construction. On the other hand, the lower bound in (7) comes from the fact that, for s  "n1/(2h 1) , a typical s-element subset of [n] becomes a Bh -set after the deletion of a small fraction of its elements. Now, the lower bound in (7) tells us that, for s  "n1/(2h 1) , the trivial upper bound Zh (n, s)  n s s is sharp up to a factor of the form (1 + o(1)) . The problem is, then, to obtain good upper bounds for Zh (n, s) for s of order n1/(2h 1) or larger, perhaps coming close to matching (6). We believe that this is possible, and put forward such a conjecture in [4], which we reproduce below for convenience. Conjecture 1.3. Fix an integer h every large enough n, we have

2 and a real number Zh (n, s)  2

✓

n sh

◆s

.

> 0. For every s

n1/(2h

1)+

and (8)

48 49

50 51

52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85

Conjecture 1.3 is proved for h = 2 in [10, 11]. The main result of this paper establishes Conjecture 1.3 for h = 3. Theorem 1.4 (Main result). For every > 0, there exists an integer n0 such that if n n1/5+  s  3n1/3 , then ⇣ n ⌘s Z3 (n, s)  3 . s

n0 and (9)

We believe that our methods for proving Theorem 1.4 can eventually be adapted to establish Conjecture 1.3 for every h, but the general h case brings considerable new difficulties, and will be addressed elsewhere. Let us compare the bounds we have for Z3 (n, s) as s varies. For s ⌧ n1/5 , Proposition 1.2(ii ) tells us that Zh (n, s) is, up to a multiplicative factor of (1 o(1))s , equal to the total number ns of selement subsets of [n]. In this range, one might therefore say that Bh -sets are ‘relatively abundant’. On the other hand, for any given > 0, for n1/5+  s ⌧ n1/3 , Theorem 1.4 and Proposition 1.2(i ) applied for h = 3 determine Z3 (n, s) up to a multiplicative factor of the form so(s) , and we see that the probability that a random s-element subset of [n] is a B3 -set is roughly of the form s (2+o(1))s . In this second range, B3 -sets are therefore scarcer. Finally, note that, by (2), if s > 3n1/3 and n is large, then Z3 (n, s) = 0. The discussion above tells us that there is a sudden change of behaviour around s0 = n1/5 . Indeed, roughly speaking, for s considerably larger than this ‘critical’ value s0 , we have that Z3 (n, s) is of s the form n/s3 o(1) ; this is in contrast to the fact that, as we have already seen, for s of smaller order than s0 , we have that Z3 (n, s) is of the form (1 o(1))s ns = (⇥(n/s))s . Theorem 1.4 implies a result in probabilistic combinatorics, which confirms the case h = 3 of a conjecture put forward in [4]. We shall discuss this corollary of Theorem 1.4 in Section 2. Notation and organization of the paper. Throughout this paper we identify a graph with the set of its edges. In particular, if G is a graph, then e 2 G means that e is an edge of G; moreover, we write both |G| and e(G) for the number of edges in G. If e = {x, y} is an edge in a graph, we sometimes write xy for e. As usual, edges are unordered pairs of vertices; however, if H is a bipartite graph with vertex classes A and B, it will be convenient to think of the edge set of H as a subset of A ⇥ B in the natural way. For a set A ⇢ V (G) we denote by e(A) = eG (A) the number of edges in the subgraph induced by A, which is denoted by G[A]. If T is a set, we denote by K(T ) the complete graph with vertex set T . For a set W ⇢ Z and x 2 Z the set W + x is defined as the set of all numbers w + x with w 2 W . We write a ⌧ b as shorthand for the statement a/b ! 0 as n ! 1. We use the standard O, o and ⇥-notation (with respect to n ! 1); the implicit constants are always absolute constants. We omit floor b c and ceiling d e symbols when they are not essential. We sometimes write a/bc for a/(bc). We are mostly interested in large n; in our statements and inequalities we often tacitly assume that n is larger than a suitably large constant. This paper is organized as follows. In Section 2 we state and prove the result in probabilistic combinatorics very briefly alluded to above. The remainder of the paper is devoted to the proof of Theorem 1.4, the structure of which is presented in Figure 1. The general approach used in the proof 3

Theorem 1.4

Lemma 4.2

Proposition 3.10

Lemma 5.1

Cl. 5.4, 5.5

Proposition 3.11

Lemma 3.3

Claim 5.13

Cor. 5.11

Claim 5.2

Key Lemmas

Lemma 4.1

Claim 5.12

Lemma 4.5

Lemma 5.7

Claim 6.2

Cl. 5.8, 5.9, 5.10

Claim 6.3 Figure 1. A diagram illustrating the flow of the proof of our main result

88

is described in Section 3. Section 4 gives some auxiliary lemmas, one of which, Lemma 4.5, plays a central technical rˆ ole. The proof of this lemma is given in Section 6. The two main propositions that together imply Theorem 1.4 (Propositions 3.10 and 3.11; see Section 3) are proved in Section 5.

89

2. Largest B3 -sets contained in random sets of integers

90

In [10, 11], the cardinality of the largest B2 -sets, i.e., Sidon sets, contained in random sets of integers was investigated. Given an integer function 0  m = m(n)  n, let us denote by [n]m an melement subset of [n] chosen uniformly at random from all such sets. Given a set R, let Fh (R) be the cardinality of the largest Bh -sets contained in R. We are interested in the random variable Fh ([n]m ). For simplicity, let us suppose m = m(n) = (1 + o(1))na for some constant 0 < a < 1. It is proved in [10, 11] that, asymptotically almost surely, that is, with probability tending to 1 as n ! 1, one

86 87

91 92 93 94 95

4

96

has F2 ([n]m ) = nb2 +o(1) , where

b2 = b2 (a) =

97 98 99 100 101

102 103 104

105

106

107 108 109 110 111 112 113 114 115 116

117 118

8 > > > :

if 0  a  1/3,

if 1/3  a  2/3,

1/3 a/2

if 2/3  a  1.

Therefore, F2 ([n]m ) undergoes a sudden change of behaviour at a = 1/3 and at 2/3. Furthermore, somewhat unexpectedly, F2 ([n]m ) does not change considerably as we vary a from 1/3 to 2/3. It is natural to ask whether a similar result holds for arbitrary h; indeed, in [4], we put forward a conjecture that states that this is the case. Theorem 1.4 implies that this conjecture holds for h = 3. Our result is as follows. Theorem 2.1 (B3 -sets contained in random sets of integers). Let 0  a  1 be a fixed constant. Suppose m = m(n) = (1 + o(1))na . There exists a constant b3 = b3 (a) such that, asymptotically almost surely, we have F3 ([n]m ) = nb3 +o(1) . (11) Furthermore,

8 > > > : a/3

if 0  a  1/5,

if 1/5  a  3/5,

(12)

if 3/5  a  1.

The piecewise linear function b3 in (12) is given in Figure 2. Proof of Theorem 2.1. We shall be somewhat sketchy in the more routine parts of the argument. We first observe that one may switch to the so called binomial model [n]p . To be more precise, let p = m/n = (1 + o(1))na 1 and put each x 2 [n] in [n]p with probability p, independently of all other elements in [n]. A standard argument tells us that it suffices to prove that F3 ([n]p ) = nb3 +o(1) p with probability 1 o(1/ m). The required lower bound for F3 ([n]p ) is established in [4]. Since F3 ([n]p )  |[n]p |, standard arguments prove Theorem 2.1 in the range a 2 [0, 1/5]. We may use Theorem 1.4 to bound the random variable F3 ([n]p ) from above, in probability, as follows. The expected number of B3 -sets of size s in [n]p is ps Z3 (n, s). For any given > 0, Theorem 1.4 implies that, for s n1/5+ , this expectation is at most ⇣ n ⌘s p 3 . (13) s p Hence, if (1 + o(1))na = pn ⌧ s3 , then this expectation is o(1/ m). In particular, for every a > 1/5, with suitably large probability, the largest B3 -sets contained in [n]p have cardinality at most max n1/5+ , na/3+

119

(10)

Since

= nb3 (a)+ . ⇤

> 0 is arbitrary, the result follows.

5

b3 (a) 1 3

1 5

1 5

1

3 5

a

Figure 2. The graph of the piecewise linear function b3 from Theorem 2.1 120

3. The proof of Theorem 1.4

121

Theorem 1.4 follows in a straightforward manner from two propositions, Propositions 3.10 and 3.11, stated at the end of this section. We need some preparations to be able to state those two propositions. The following definition introduces a central object in the proof.

122 123

124 125 126

127 128

129 130 131

132 133 134 135 136

137 138

139

Definition 3.1 (Collision graph CT ). Given a set T ⇢ [n], we define the collision graph CT on the vertex set [n] by letting {a, b} with a, b 2 [n] and a 6= b be an edge whenever there exist z1 , z2 , z3 , z4 2 T such that a + z1 + z2 = b + z3 + z4 . (14) Proposition 3.2. Suppose that S ⇢ [n] is a B3 -set. Then for every T ⇢ S, the set S \ T is an independent set in CT . Proof. Suppose on the contrary that a, b 2 S \ T with a 6= b satisfies (14) with z1 , z2 , z3 , z4 2 T . From the fact that S is a B3 -set we deduce that the multisets {a, z1 , z2 } and {b, z3 , z4 } coincide. Since a 2 S \ T and z3 and z4 2 T , we obtain that a = b, which is a contradiction. ⇤ In view of Proposition 3.2, our general strategy for estimating the number of B3 -sets of a given size s will be as follows: we first enumerate seed B3 -sets T with |T | ⌧ s and then we bound the number of independent sets in CT for each such T . The following lemma, which is implicit in the work of Kleitman and Winston [9] (see also [11, Lemma 3.1]), will be used to bound the number of independent sets. Lemma 3.3. Let G be a graph on N vertices, let q be an integer and let 0  numbers with R e q N. Suppose e(A)

✓

|A| 2

◆

for any A ⇢ V (G) with |A| 6

R.

 1 and R be real (15) (16)

140

Then, for all integers m

141

When applying Lemma 3.3, we shall often take R = N = |V (G)| for some number Hypothesis (15) then becomes e q >1

142

143

0, the number of independent sets in G of cardinality q + m is at most ✓ ◆✓ ◆ N R . (17) q m

and the bound (17) becomes

> 0. (18)

151

◆✓ ◆ |V (G)| |V (G)| . (19) q m In order to prove our main result, Theorem 1.4, we shall enumerate all possible seed sets T and show that the corresponding graphs CT are quite dense. In fact, they are dense enough that we can apply Lemma 3.3 to establish that the number of extensions of T to a significantly larger B3 -set S is rather small. More precisely, for every B3 -set T we are interested in proving lower bounds for eCT (A) for arbitrary but somewhat large A ⇢ [n]. It turns out that we shall need to consider two separate cases, depending on the structure of T . We now need some definitions. Let G be a graph on the vertex set T ⇢ [n] and let z 2 [n] be arbitrary. Denote by G2 = G ⇥ G the Cartesian product of the edge set of G with itself.

152

Definition 3.4 (Representation count RG ). Let

144 145 146 147 148 149 150

RG (z) = 153 154 155 156 157 158 159 160 161

✓

(z1 z2 , z3 z4 ) 2 G2 : z = (z1 + z2 )

Definition 3.5 (Collision multigraph CeG ). Let CeG be the multigraph with vertex set [n] in which the multiplicity of each {a, b} 2 [n] a). 2 is exactly RG (b The reason we introduce this multigraph version of CT is that it will be easier to estimate from below the number of multi-edges that are induced by subsets A ⇢ [n]. We can then establish bounds for CT through the following proposition. Proposition 3.6. For every non-empty graph G ⇢ z2[ n,n]

163 164 165 166 167

(20)

Clearly RG ( z) = RG (z) for every z. In what follows, T will always be a B3 -set, and hence we shall always have RG (0) = 0. Finally, we mention that we shall only be interested in RG (z) for z 2 { n + 1, . . . , 1} [ {1, . . . , n 1} ⇢ [ n, n].

T 2

eCT (A) max RG (z) 162

(z3 + z4 ), all zi s distinct .

and A ⇢ [n] we have eCeG (A).

(21)

Proof. Note that {a, b} 2 CeG [A] implies that RG (b a) 1 which means that b a = (z1 + z2 ) (z3 + z4 ) for some zi 2 T , and thus {a, b} 2 CT [A] (see Definition 3.1). The proposition follows. ⇤ A substantial part of this paper is dedicated to proving the existence of suitable graphs G for which we can bound maxz2[ n,n] RG (z) and then apply Proposition 3.6. Remark 3.7. Note that, in the definition of RG (z), the elements z1 , z2 , z3 and z4 are required to be all distinct. This restriction allows us to avoid the “degenerate” case in which z = z1 z3 7

168 169 170 171 172

for z1 , z3 2 T . In this case, for every x 2 NG (z1 ) \ NG (z3 ) we have z = (z1 + x) (z3 + x) with (z1 x, z3 x) 2 G2 , which means that, without the restriction, the value of RG (z) could be as large as |NG (z1 ) \ NG (z3 )|, which, in turn, could be as large as |T | 2. We now define a quantity QG that will help us bound maxz2[

n,n] RG (z).

Definition 3.8 (Moment generating function of RG ). Let X QG = exp RG (z).

(22)

z2[n]

173

As already observed, RG vanishes at 0 for any B3 -set T and is an even function. Thus max RG (z) = max RG (z)  log QG .

z2[ n,n] 174 175 176 177

In view of the definitions above and Proposition 3.6 and Lemma 3.3, our goal is to enumerate B3 sets T ⇢ [n] and graphs G ⇢ T2 such that log QG is not too large, while at the same time eCeG (A) is large for every “large enough” set A ⇢ [n]. This discussion motivates our next definition. Definition 3.9 (Bounded set). Let ⇠=

178 179 180

181

(23)

z2[n]

1 106 log n

and

↵i = ⇠ 2

i+1

1

for i

0,

(24)

and let " > 0 be a fixed constant. Given 1 and a non-negative integer i, a set T is said to satisfy P ,",i if it is a B3 -set and there exists a graph Gi on the vertex set T such that ✓ ◆ |T | (a) e(Gi ) 1 ↵i , 2 ✓ ◆ |T | X 1 i" (b) QGi  en exp n . j j=1

182 183 184

A set T is called ( , ")-bounded if it satisfies P

186 187

188 189 190 191 192 193

for i = 0, 1, . . . , d1/"e.

We now summarize our strategy for proving Theorem 1.4. Fix a positive constant may and shall suppose that  1. Let " = /13 and note that 1 1  + 5 25" 5

185

,",i

and

3

13" = 3

3

12".

> 0. We

(25)

Let s be an integer satisfying the assumptions of the theorem. In particular, s n1/(5 25") by our choice of ". Our goal is to estimate the number of B3 -sets of cardinality s. For the remainder of the paper, we let s5 25" = (s) = 1. (26) n We classify the B3 -sets of size s into two types, depending on whether or not the cardinality of their largest ( , ")-bounded subsets is greater than s1 6" . We shall prove the following two propositions, estimating the number of B3 -sets of cardinality s of each type separately. These two propositions together easily imply Theorem 1.4. Proposition 3.10. Let " > 0, let n be a sufficiently large integer, and let s 2 [n1/(5 25") , 3n1/3 ] be a given integer. Let be as defined in (26). The number of B3 -sets of cardinality s contained 8

194

195 196 197

in [n] that contain a ( , ")-bounded set larger than s1 ✓ ◆s n . s3 12" o(1)

6"

is at most (27)

Proposition 3.11. Let " > 0, let n be a sufficiently large integer, and let s 2 [n1/(5 25") , 3n1/3 ] be a given integer. Let be as defined in (26). The number of B3 -sets of cardinality s contained in [n] that do not contain any ( , ")-bounded set larger than s1 6" is at most ✓ ◆s n . (28) s3 8" o(1)

215

Before we proceed with the formal proofs, let us briefly discuss our general approach. Every B3 set with s elements that contains a ( , ")-bounded set with at least s1 6" elements will be shown to contain a set T with |T | = s1 6" which satisfies P100 ,",0 (see Lemma 4.2). Using Lemma 3.3, we shall be able bound the number of possible extensions of any such set T to a B3 -set with s elements. This is because the graph CT will be shown to satisfy an appropriate local density condition (see Lemma 5.1). Showing this is the main difficulty in this part of the argument. The details are given in Section 5.1. The proof of Proposition 3.11 is somewhat more complicated. First we show that any B3 -set of cardinality s must contain a ( , ")-bounded subset of size at least s1/7 (see Lemma 4.1). In particular, every such B3 -set contains a maximal ( , ")-bounded subset with at least s1/7 elements. Our strategy will therefore be to estimate, for each B3 -set T with |T | < s1 6" , the number of B3 sets S such that T ⇢ S and T is a maximal ( , ")-bounded subset of S. The maximality of T will be shown to imply that the set of elements that can appear in S \ T admits a certain structure (see Definition 5.6 and Lemma 5.7). More concretely, we shall show that S \ T ⇢ Te for some set Te ⇢ [n] such that the graph CT [Te] satisfies certain local density conditions that allow us to use Lemma 3.3 to bound the total number of such possible extensions S of T appropriately (the precise local density condition is given in Corollary 5.13) The remainder of the paper is devoted to proving Propositions 3.10 and 3.11.

216

4. Auxiliary lemmas

217

We now give three auxiliary lemmas. The two lemmas in Section 4.1 are quite simple, while the lemma given in Section 4.2, Lemma 4.5, is somewhat more technical. However, Lemma 4.2 will be one of the key lemmas that will allow us to prove local density results for certain induced subgraphs of the collision graph CT .

198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214

218 219 220 221 222 223 224 225 226

4.1. Bounded sets. Our first lemma states that for any 1 and any " > 0, every B3 -set S contains a ( , ")-bounded subset whose size is at least a small power of |S|. Lemma 4.1. For any cardinality |T | |S|1/7 .

1, " > 0, and B3 -set S ⇢ [n] there exists a ( , ")-bounded set T ⇢ S of

Proof. Observe that S contains a B4 -set with d|S|1/7 e elements. Indeed, one may construct such set greedily by starting from an empty set and sequentially adding to it elements of S. As long as 9

227 228 229 230 231

the constructed set T has fewer than |S|1/7 elements, one can always add to T an arbitrary element from the (non-empty) set S \ (4T 3T ), which assures that T remains a B4 -set. Hence, we may choose a B4 -set T ⇢ S with |T | = d|S|1/7 e. Let G be the complete graph on the vertex set T . The fact that T is a B4 -set implies that RG (z) 2 {0, 1} for every z. Indeed, if some (z1 z2 , z3 z4 ), (z10 z20 , z30 z40 ) 2 G2 , with |{z1 , z2 , z3 , z4 }| = 4 and |{z10 , z20 , z30 , z40 }| = 4, satisfy (z1 + z2 )

232

(z3 + z4 ) = z = (z10 + z20 )

(z30 + z40 ),

(29)

then z1 + z2 + z30 + z40 = z10 + z20 + z3 + z4 .

233

(30)

Since T is a B4 -set, we must have, as multisets, {z1 , z2 , z30 , z40 } = {z10 , z20 , z3 , z4 }.

234 235

(31)

This forces {z1 , z2 } = {z10 , z20 } and {z3 , z4 } = {z30 , z40 }. Consequently RG (z)  1. In particular, QG is a sum of n terms which are either e0 = 1 or e1 = e. Therefore, QG  en.

236 237 238 239 240 241 242 243 244 245

246

247 248

Clearly, Gi = G = K(T ) satisfies both (a) and (b) of Definition 3.9 for any i Hence, T is a ( , ")-bounded set.

0,

1, and " > 0. ⇤

The second lemma allows one to pass to subsets of a convenient cardinality when dealing with ( , ")-bounded sets. Moreover, we shall see that we may carry out this procedure without significantly a↵ecting the “boundedness” parameters. Lemma 4.2. Let 1 and an integer i 0 be given and suppose that T ⇢ [n] satisfies P ,",i . 1/100 For every m satisfying n  m  |T |, there exists T 0 ⇢ T with |T 0 | = m such that T 0 satisfies P100 ,",i . Proof. Let Gi be a graph whose existence is asserted in the definition of P ,",i . A simple averaging argument shows that there exists a T 0 ⇢ T with |T 0 | = m such that ✓ ◆✓ ◆ ✓ ◆✓ ◆ 1 |T | 2 |T | 1 m |T | 0 eGi (T ) e(Gi ) = e(Gi ) . (33) m 2 m 2 2 Taking G0i = Gi [T 0 ] and recalling that Gi satisfies (a) of Definition 3.9 yields ✓ ◆✓ ◆✓ ◆ 1 |T | m |T | 0 e(Gi ) 1 ↵i 2 2 2 ✓ ◆ m = 1 ↵i . 2

(34)

Using the facts that n1/100 < m  |T |  3n1/3 , that Gi satisfies (b) of Definition 3.9, and well-known estimates for the harmonic numbers, we obtain, for every large enough n, ◆ ✓ ✓ |T | |T 0 | ◆ X X 1 1 i" i"  en exp 100 n . (35) QG0i  QGi  en exp n j j j=1

249

(32)

It follows from (34) and (35) that T 0 satisfies P100 10

j=1

,",i .

⇤

250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270

271

4.2. A technical lemma on the local density of CT . We now state a key technical lemma that will help us give lower bounds for the local density of the collision graph CT . We need the following definition. Definition 4.3 (Bipartite graph HT (A, B)). Given a set T ⇢ [n], we define the graph HT on the vertex set ([n] ⇥ {1}) [ ([2n] ⇥ {2}) by letting {(a, 1), (b, 2)} be an edge whenever b a 2 T . For A ⇢ [n] and B ⇢ [2n], we denote by HT (A, B) the subgraph of HT induced by (A ⇥ {1}) [ (B ⇥ {2}). Remark 4.4. In the definition above, we wish to have the disjoint union of [n] and [2n] as the vertex set of HT . The standard way of producing such a disjoint union involves the use of the Cartesian product, as above. In what follows, we shall be less formal and we shall refer to vertices as a 2 A, b 2 B, etc, instead of (a, 1) 2 A ⇥ {1}, (b, 2) 2 B ⇥ {2}, etc. The following technical lemma allows us to obtain a lower bound on eCT (A) in terms of the edge-density of the graph HT (A, B) for every sufficiently large B3 -set T that satisfies P ,",i . When applying this lemma, we have to come up with a suitable set B (see the proofs of Claim 5.2 and Corollary 5.11). Recall that ⇠ and the ↵i are defined in 24. Lemma 4.5. The following holds for every integer i 0, and every " > 0, 1, D 5000 2 1/100 and 2 (0, 1] satisfying ↵i /100⇠. Suppose that a set T ⇢ [n] with at least n elements satisfies P ,",i . Moreover, suppose that A ⇢ [n] and B ⇢ [2n] are such that the graph H = HT (A, B) satisfies (I) every vertex of A has degree at least |T |; (II) the average degree of the vertices in B is D. Then

✓ eCT (A) = ⌦

2 |A| D 2 |T |2

ni" (log n)3

◆

.

(36)

The proof of Lemma 4.5 will be given in Section 6.

272

5. Proofs of Propositions 3.10 and 3.11

273

5.1. Sets containing a large bounded subset. Let us now prove Proposition 3.10, which deals with the case in which the “seed” set contains a ( , ")-bounded set of cardinality greater than n1 6" . Our main tools will be Lemma 3.3 and the following estimate on the number of edges induced by small sets of vertices in the collision graph CT when T is a bounded set.

274 275 276 277 278

279 280 281

Lemma 5.1. There exists an absolute constant C > 0 such that the following holds. If T satisfies P100 ,",0 and |T | n1/100 , then for any A ⇢ [n] with |A| (C/|T |2 )n, we have ✓ ◆ |A|2 |T |4 eCT (A) = ⌦ . (37) n(log n)3 We give the proof of Proposition 3.10 before proving Lemma 5.1. Proof of Proposition 3.10. We wish to estimate the number of B3 -sets S of size s that contain a ( , ")-bounded subset with more than s1 6" elements. Suppose that T ⇢ S is a ( , ")-bounded 11

282 283 284

set with |T | s1 6" . By Definition 3.9, T must satisfy P ,",i for i = 0. By Lemma 4.2, we may assume without loss of generality that the cardinality of T is exactly s1 6" and T satisfies P100 ,",0 . Lemma 5.1 then implies that CT satisfies (16) with R= n=

285

286

and

Take

✓ =⌦

|T |4 n(log n)3

◆

(26)

= ⌦

✓

Cn Cn = 2 12" 2 |T | s (s1

s5

6" )4

25" (log n)3

q = (log n)/ = O (log n)4 s1 287 288

289

290

291 292 293 294

◆ "

(38) ✓ =⌦

= o(s).

297

298

299 300 301 302

(40)

By the discussion above, this completes the proof of Proposition 3.10.

⇤

It now remains to prove Lemma 5.1. Proof of Lemma 5.1. Let C = 109 . We shall show that this choice of C will do. Suppose that T satisfies P100 ,",0 and let A ⇢ [n] with |A| Cn/|T |2 be arbitrary. Recall that, by definition, there exists a graph G0 ⇢ T2 satisfying (a) and (b) of Definition 3.9 with i = 0 and replaced by 100 . Our goal is to establish a lower bound on eCeG (A) and then apply Proposition 3.6 to obtain the lemma. Let H = HT (A, [2n]) (recall Definition 4.3). Let c = 10 W = w 2 [2n] : degH (w)

296

(39)

Note that (18) is satisfied. Hence, Proposition 3.2 and Lemma 3.3 yield that the number of B3 -sets of cardinality s that contain a set T satisfying P100 ,",0 with cardinality s1 6" is at most ✓ ◆✓ ◆✓ ◆ ✓ ◆s q s1 6" n n Cn/s2 12" Ce s n s1 6" q s q s1 6" s2 12" (s q s1 6" ) ⇢✓ ◆(s q s1 6" )/s s ⇢ ✓ ◆1 o(1) s Ce Ce s (41) =n = n s2 12" (1 o(1))s (1 o(1))s3 12" ✓ ◆s n = . s3 12" o(1)

0

295

◆ 1 . s1 " (log n)3

4

and let

c |T |2 |A|/n

(42)

(the vertices in W have very large degree). Notice that, since |H|  |A| |T |, we have |W |  n/(c|T |). Also, set A0 = {a 2 A : |NH (a) \ W | < 0.1 |T |}. (43) Claim 5.2. If |A0 |  |A|/2, then the conclusion of the lemma holds. More precisely, ✓ 2 2 4◆ |A| |T | 0 eCT (A \ A ) = ⌦ . n(log n)3

(44)

Proof. We apply Lemma 4.5 to the graph H 0 = HT (A \ A0 , W ) ⇢ H with i = 0, B = W , 100 in place of , and A \ A0 in place of A. We proceed in steps. (i ) Set = 0.1 and notice that lemma.

2

= 0.01 = ↵0 /100⇠, and thus

12

satisfies the condition of the

303 304 305 306 307

(ii ) We assumed that T satisfies P100 ,",0 and that |T | n1/100 , and therefore T satisfies the conditions of the lemma, with 100 in place of . (iii ) From the definition of A0 in (43), it follows that degH (a) = |NH (a) \ W | |T | for all 0 a2A\A. (iv ) Finally, the average degree of the vertices in W is D=

308 309

310

311

312 313 314 315

As C = 109 , c = 10

4

|H 0 | |W |

and |A|

c |A| |T |2 . 20n

|A| 0.1 |T | · 2 |W |

Cn/|T |2 , we have D

cC/20 = 5000.

From Lemma 4.5, with D = ⌦(|A| |T |2 /n) as in (45) and |A| Cn/|T |2 , we conclude that ✓ ◆ ✓ 2 2 4◆ 3 6 |A| |T | 0 2 |A| |T | eCT (A \ A ) = ⌦ =⌦ , 2 3 n (log n) n(log n)3 as required. In view of Claim 5.2, let us assume that |A0 |

(45)

(46) ⇤

|A|/2.

Definition 5.3 (Auxiliary graph Aux). Let Aux be a bipartite graph with classes consisting of A0 and a disjoint copy of [3n] as follows. For x 2 A0 , the neighbors of x in Aux are all elements of the form y = x + z1 + z2 for some z1 z2 2 G0 such that x + z1 , x + z2 2 / W . In other words, z1 z2 2 G0 [T \ (W x)].

317

Suppose that a pair of distinct x, x0 2 A0 is connected by a path of length two in Aux. We classify this path as follows:

318

non-degenerate path: if the path is of the form

316

x, x + z1 + z2 = x0 + z3 + z4 , x0 319

with (z1 z2 , z3 z4 ) 2 G20 and zi s distinct;

degenerate path: if the path is of the form x, x + z1 + z = x0 + z2 + z, z 0

320 321 322 323 324

326 327 328 329 330

(47)

Note that the two cases above are exhaustive since elements in an edge of G0 are necessarily distinct (i.e., G0 has no loops). Denote by d(x, x0 ) the number of degenerate paths between x and x0 and by p(x, x0 ) the total number of 2-paths connecting them. Note that a non-degenerate path between x, x0 corresponds to an ordered pair of edges of G0 counted by RG0 (x0 x) = RG0 (x x0 ) (see (20)). Therefore, RG0 (x

325

with (z1 z, z2 z) 2 G20 .

x0 )

p(x, x0 )

d(x, x0 ).

(48)

(We have an inequality instead of equality in (48) above because, owing to the definition of Aux, the first edge of the pair must come from G0 [T \ (W x)] and the second edge of the pair from G0 [T \(W x0 )], and hence not all pairs counted by RG0 (x x0 ) yields appropriate 2-paths in Aux.) P In order to estimate eCeG (A0 ) = x,x0 2A0 RG0 (x x0 ) we bound the number of degenerate paths 0 P and estimate the total number of 2-paths. Here and in what follows, we write x,x0 2A0 for the sum over all unordered pairs {x, x0 } ⇢ A (x 6= x0 ). 13

x0

x

x A0

x0 HT (A, [2n] \ W )

Aux x + z 1 + z = x 0 + z2 + z

[3n]

x + z 1 = x 0 + z2

Figure 3. Owing to the definition of Aux, we know that x + z1 = x0 + z2 does not belong to W . Consequently, (x, x + z1 , x0 ) is a two-path in HT (A, [2n] \ W ), and hence a member of P. 331 332 333 334 335

P Let D denote the set of all degenerate paths in Aux, so that |D| = x,x0 2A0 d(x, x0 ). Also, let P be the set of all paths of length two in HT (A0 , [2n] \ W ) with both endpoints in A0 . We will provide an upper bound for |D| by defining a map : D ! P, estimating |P| and bounding | 1 (P )| for all P 2 P. Claim 5.4. We have |D| 

336 337 338 339 340 341 342 343 344 345

c |A|2 |T |4 . n

Proof. First we define a map : D ! P as follows. For a degenerate path in Aux between x, x0 2 A0 as in (47), we infer that x, x0 are connected by a path of length two in H (i.e., x, x + z1 = x0 + z2 , x0 ). Given the definition of Aux, we also know that x+z1 = x0 +z2 2 / W . Let map the degenerate 0 0 0 0 path x, x + z1 + z = x + z2 + z, x to the path x, x + z1 = x + z2 , x , which indeed belongs to P—see Figure 3. Since there are at most |T | choices for z 2 T such that both {z1 , z} and {z2 , z} are edges in G0 , we conclude that | 1 (P )|  |T | for any P 2 P. Hence |D|  |T | |P|. The cardinality of P can be bounded from above by |A0 | |T | · c |T |2 |A|/n. Indeed, there are at most |A0 | |T | choices for the first edge of the path and since the path’s middle vertex, which is determined by the first edge, is not in W , it follows from (42) that the number of choices for the second edge is at most c |T |2 |A|/n. Consequently, |D|  |T | |P|  |T | · |A0 | |T | ·

346 347

348 349

c |T |2 |A| c |A|2 |T |4  . n n

Claim 5.5. The number of 2-paths between vertices of A0 in Aux, namely the sum is at least |A|2 |T |4 . 7 ⇥ 64n

⇤ P

x,x0 2A0

p(x, x0 ), (49)

Proof. The total number of 2-paths between pairs of vertices of A0 in Aux can be bounded below by using Jensen’s inequality: ◆ ✓ ◆ X ✓deg |Aux|/3n |Aux|2 Aux (y) 3n . (50) 2 2 7n y2[3n]

350 351 352

The claim will follow after we obtain a lower bound for |Aux|. Note that by construction (see Definition 5.3) and the fact that T is a B3 -set, the degree of any x 2 A0 in Aux is precisely eG0 (T \ (W x)). Since |NH (x) \ W | < 0.1 |T | and NH (x) = 14

353

T + x ⇢ [2n], we have |T \ (W

x)| = |T | = |T |

354

355 356

357

358

x)| = |T |

Since x 2 A0 was arbitrary, we have |Aux| Therefore X p(x, x0 ) Since |A0 |

|(T + x) \ W |

(51)

|NH (x) \ W | > 0.9 |T |.

Since G0 satisfies (a) with i = 0, we must have ✓ ◆ |T \ (W x)| degAux (x) = eG0 T \ (W x) 2

x,x0 2A0

|K(T ) \ G0 |

✓

0.9 |T | 2

◆

↵0

✓

|T | 2

◆

|T |2 . 4 (52)

(|A0 | |T |2 /4)2 . 7n

|Aux|2 7n

(53) ⇤

It follows from Claims 5.4 and 5.5, together with (48) and c = 1/10000, that

0

eCeG (A0 ) 0

X

(48)

p(x, x0 )

d(x, x0 )

Cl. 5.4 & 5.5

x,x0 2A0

Since QG0 satisfies (b) with i = 0, and 100

|A|2 |T |4 . 500n

(54)

1, we have

QG0  en exp 100 (1 + log |T |)  exp(200 log n). 360

>

|A0 ||T |2 /4.

|A|/2, the claim follows.

eCeG (A)

359

|T \ (W

(55)

It follows by (23) and Proposition 3.6 that eC (A)

eCeG (A) 0

log QG0

eCeG (A) 0

200 log n

|A|2 |T |4 . 105 n log n

(56) ⇤

361

Hence Lemma 5.1 is proved.

362

368

5.2. Sets not containing a large bounded subset. We now turn to the proof of Proposition 3.11, that is, we enumerate B3 -sets such that all of its ( , ")-bounded subsets have fewer than s1 6" elements. We shall do this by bounding, for any given ( , ")-bounded set T , the number of ways one can extend T to a B3 -set S in such a way that T remains a maximal ( , ")-bounded subset of S. In what follows, we show that extensions preserving a ( , ")-bounded set T as maximal must admit certain structural properties that severely restrict the number of possible extensions.

369

Definition 5.6. Given a ( , ")-bounded set T , let

363 364 365 366 367

370

Te = x 2 [n] : T [ {x} is a B3 -set but not a ( , ")-bounded set .

(57)

Also, for i 2 {0, 1, . . . , d1/"e}, let

Tei = x 2 Te : i is the smallest index such that T [ {x} does not satisfy P 15

,",i

.

(58)

371 372

373 374 375 376 377 378

379

380 381 382 383

384

Note that, by definition, if a B3 -set S contains T and T is a maximal ( , ")-bounded subset of S, then S \ T ⇢ Te. Note that, clearly, the sets Tei partition Te and Te =

d1/"e

[

i=0

Tei .

(59)

The next lemma gives us important information on the sets Tei . The sets Bi , whose existence is asserted in this lemma, will be crucial for us to prove that CT [Tei ] satisfies a local density condition, as specified in Corollary 5.11. The Bi will be used in an application of Lemma 4.5 in the proof of Corollary 5.11. Lemma 5.7. Let i 2 {0, 1, . . . , d1/"e} and suppose that a set T satisfies P set Bi = Bi (T ) ⇢ [2n] with e2 (e + 1)|T |4 |Bi | < ni·" such that, for every x 2 Tei , |(T + x) \ Bi | ↵i |T |.

,",i .

There exists a (60) (61)

Proof. Since T satisfies P ,",i , there exists a graph Gi on the vertex set T that satisfies (a) and (b) of Definition 3.9. Let us fix such a graph Gi for the remainder of the proof of Lemma 5.7. For technical reasons, it will be convenient to introduce the following definition: for each w 2 [2n] and z 2 [n], set 8 <1 if z = ±(w a b) for some {a, b} 2 G , i fi (w, z) = (62) :0 otherwise. Also, for each w 2 [2n], let

Ui,w =

X

(exp RGi (z)) e2 fi (w, z).

(63)

z2[n] 385

In what follows, we will show that the set Bi defined by ⇢ ni" QGi Bi = w 2 [2n] : Ui,w > (e + 1)|T |(|T | + 1)

386

satisfies the conclusions of the lemma.

387

Claim 5.8. We have |Bi | < e2 (e + 1)|T |4 /( ni·" ).

388

Proof. We start by proving the following inequality, which will be used shortly: X for any z 2 [n], we have fi (w, z)  2 e(Gi ).

(64)

(65)

w2[2n]

389 390

This inequality holds since each edge {a, b} of Gi may only contribute to the sum on the left hand side with the two entries fi (a + b + z, z) and fi (a + b z, z). Now observe that X X X Ui,w = (exp RGi (z)) e2 fi (w, z) w2[2n]

w2[2n] z2[n]

= e2

X

z2[n]

exp RGi (z)

X

w2[2n]

fi (w, z)  e2 QGi 2 e(Gi ) < e2 |T |(|T | 16

1)QGi . (66)

391

On the other hand,

X

Ui,w

393 394 395

396

|Bi |

Ui,w

w2Bi

w2[2n] 392

X

ni·" QGi , (e + 1)|T |(|T | + 1)

⇤

which implies (60), concluding the proof of the claim.

It remains to prove that for every x 2 Tei , we have |(T + x) \ Bi | ↵i |T |. Fix an arbitrary x 2 Tei . For y 2 T , denote by Gi [ {xy} the graph with vertex set V (Gi ) [ {x} and edge set E(Gi ) [ {xy}. Let Di,xy = QGi [{xy} QGi . (68) Expanding, we obtain Di,xy =

X

exp RGi (z)

z2[n]

n

⇣ exp RGi [{xy} (z) | {z

397

The following claim relates Di,xy and Ui,x+y .

398

Claim 5.9. We have

RGi (z)

(‡)

⌘

Di,xy  Ui,x+y . 399 400 401 402

403 404 405 406 407 408

410

413

(70)

(72)

In particular, the term (‡) in (69) is 0, e 1 or e2 1. To summarize, regardless of whether fi (w, z) is 0 or 1, we have

Therefore, Di,xy 

412

(69)

or due to a pair (z1 z2 , xy) such that z = (z1 + z2 ) w, where, in both cases, we require {x, y} \ {z1 , z2 } = ;. If fi (w, z) = 0 then there are no such pairs and we must have RGi [{xy} (z) = RGi (z). In this case, the term (‡) in (69) is 0. Since T is a B3 -set, there can be at most one edge {a, b} 2 Gi such that z = w a b and at most one edge {a0 , b0 } 2 Gi for which z = (x + y) a0 b0 . Therefore, we always have RGi [{xy} (z)  RGi (z) + 2. Consequently, in this case

(‡)  e2 fi (w, z). 411

o 1 . }

Proof. Let w = x + y. We shall prove the claim by showing that every term in the sum (69) is bounded above by its corresponding term in the sum (63) defining Ui,w . Let z 2 [n] be arbitrary. Note that any di↵erence between RGi [{xy} (z) and RGi (z) must be either due to a pair (xy, z1 z2 ), z1 z2 2 Gi , satisfying z = (x + y) (z1 + z2 ) = w (z1 + z2 ), (71)

RGi (z)  RGi [{xy} (z)  RGi (z) + 2. 409

(67)

X

(exp RGi (z)) e2 fi (w, z) = Ui,w = Ui,x+y .

(73) ⇤

z2[n]

Next we show that the e↵ect in the moment function caused by adding multiple edges incident to x to the graph Gi is essentially the sum of the e↵ects of each edge xy being added.

17

414

Claim 5.10. For any Y ⇢ T , letting G0i = Gi [ {xy : y 2 Y }, we have X QG0i QGi  (e + 1) Di,xy .

(74)

y2Y

415 416 417

Proof. Since G0i \Gi = {xy : y 2 Y } contains only edges incident to x, the di↵erence RG0i (z) RGi (z) equals the number of solutions to z = ±(x + y a b) for some y 2 Y and {a, b} 2 Gi with {x, y} \ {a, b} = ;. Let us bound the number of such solutions. To this end, suppose that z =x+y

418 419 420 421 422

423 424

a

b = x + y0

a0

b0

(75)

for y, y 0 2 Y and {a, b}, {a0 , b0 } 2 Gi with {x, y} \ {a, b} = ; and {x, y 0 } \ {a0 , b0 } = ;. Then y+a0 +b0 = y 0 +a+b and, since these elements come from the B3 -set T , we conclude that {y, a0 , b0 } = {y 0 , a, b}. It follows that y = y 0 and {a, b} = {a0 , b0 }. Hence, there is at most one solution to z = x+y a b and at most one solution to z = x+y a b. Consequently, RG0i (z) RGi (z)  2 and exp RGi (z)  (exp RGi (z)) (e2 1). (76) z := exp RG0i (z) Moreover, RG0i (z) > RGi (z) only if RGi [{xy⇤ } (z) > RGi (z) for some y ⇤ 2 Y , and, therefore, in that case, we have z

 (e + 1) · exp RGi (z) · (e

1)

= (e + 1) exp RGi (z) + 1

exp RGi (z)

 (e + 1) exp RGi [{xy⇤ } (z) exp RGi (z) X  (e + 1) · exp RGi [{xy} (z) exp RGi (z) .

(77)

y2Y

425 426

Note that if RG0i (z) = RGi (z), then both the left-hand and the right-hand side of (77) are zero. In other words, X exp RGi [{xy} (z) exp RGi (z) (78) z  (e + 1) y2Y

427

holds for all z 2 [n]. Consequently, X QG0i QGi =

z

z2[n]



X

(e + 1)

z2[n]

= (e + 1)

X

X

y2Y

exp RGi [{xy} (z)

exp RGi (z) ⇤

Di,xy .

y2Y 428

Setting Y = {y 2 T : x + y 2 [2n] \ Bi } 18

(79)

429

in Claim 5.10 yields that G0i = Gi [ {xy : y 2 Y } satisfies X QG0i  QGi + (e + 1) Di,xy y2Y

(70)



(64)+(79)

 

Def. 3.9(b)

430 431 432

433

434

QGi + (e + 1)

X

QGi + (e + 1) ✓

QG i 1 +



en exp



en exp

X

ni" QGi (e + 1)|T |(|T | + 1) y2Y ◆ i"

n |T | + 1

✓

n

✓

ni"

i"

◆ |T | X 1 j=1

j

437 438

440 441

◆

X 1◆ , j

which means that satisfies (b) of Definition 3.9 with T [ {x} in place of T . Since our assumption that x 2 Tei implies that T [ {x} does not satisfy P ,",i , the graph G0i must fail (a) of Definition 3.9. Thus ✓ ◆ |T | + 1 0 e(Gi ) = e(Gi ) + |Y | < 1 ↵i (81) 2 and, as Gi satisfies (a), we conclude that ⇢✓ ◆ |T | + 1 |Y | < 1 ↵i 2

✓

|T | 2

◆

= 1

↵i |T |.

(82)

From the definition of Y in (79) and the fact that T ⇢ [n], x 2 [n], it follows that x).

(83)

Hence x)| = |T |

|Y |

Since x 2 Tei was arbitrary, the proof of Lemma 5.7 is complete.

↵i |T |.

(84) ⇤

Recall Definition 4.3 from Section 4.2. Lemma 5.7 implies that for every i 2 {0, 1, . . . , d1/"e}, there exists a B = Bi with |B| = O(|T |4 /( ni" )) such that for every A ⇢ Tei , |HT (A, B)|

439

ni" exp |T | + 1

j=1

G0i

|(T + x) \ Bi | = |T \ (Bi

436

✓

(80)

|T |+1

Y = T \ (Bi 435

Ui,x+y

y2Y

↵i |T | |A|.

(85)

Together with Lemma 4.5, this yields the following corollary. Corollary 5.11. Suppose that T is a ( , ")-bounded set with cardinality at least n1/100 and less than s1 6" . For every i 2 {0, . . . , d1/"e 1} and any set A ⇢ Tei with |A|

s

19

2+8"

n,

(86)

442

we have eCT (A) = ⌦

443 444

445 446

447 448 449

450 451

452 453

✓

◆ ↵i3 |A|2 . |T | n" (log n)3

(87)

Proof. Fix i 2 {0, . . . , d1/"e 1} and let B = Bi (T ) be the set from Lemma 5.7. In particular, |B| = O |T |4 /( ni" ) . Also let = ↵i . (88) We now show that the the graph H = HT (A, B) satisfies all the conditions of Lemma 4.5 with i + 1 in place of i. We proceed in steps. From (24), we have ↵i+1 ↵i+1 i+1 1) i+2 2 2 = ↵i2 = ⇠ 2(2 = ⇠2 = > . (89) ⇠ 100⇠ Our assumptions that i < d1/"e and that T is ( , ")-bounded imply that T satisfies P ,",i+1 (see Definition 3.9). Moreover, we also assume that |T | n1/100 . Lemma 5.7 implies that every a 2 A ⇢ Tei satisfies degH (a) = |(T + a) \ B| |T |. (90)

Finally, recalling that = s5 25" /n 1 (see (26)) and that s n1/(5 25") , we have that the average degree D of the vertices in B satisfies ✓ ◆ ✓ 3 17" i" ◆ |H| |A||T | s 2+8" n |T | s 2+8" n ni" s n D= =⌦ =⌦ s" 5000, (91) 3 3 |B| |B| |B| |T | |T | where we used that |T | < s1 have

6"

and that n is large. By Lemma 4.5 (with i + 1 in place of i), we

✓ eCT (A) = ⌦

|A| |T |2 n(i+1)" (log n)3

✓

|H| |B|

◆2 ◆

◆ |A| |T |2 |T | |A| · D |B| n(i+1)" (log n)3 ✓ ◆ D |A|2 |T |3 = ⌦ 3 |B| n(i+1)" (log n)3 ✓ ◆ D |T |4 (88) 2 = ⌦ ↵i3 |A| . |T | n" (log n)3 |B| ni" (91)

= ⌦

454 455

456 457 458 459 460 461

462

✓

2

2

(92)

Since |B| = O |T |4 /( ni" ) , the term |T |4 /|B| ni" on the right hand side of (92) can be replaced by 1. Hence, from (91) and (92) it follows that (87) holds and the corollary is proved. ⇤ Let s 2 [n1/(5 25") , 3n1/3 ] be fixed and t < s1 6" be an integer. In order to prove Proposition 3.11, we will estimate how many B3 -sets have a maximal ( , ")-bounded set T with cardinality t. As we observed above, if T is a maximal ( , ")-bounded subset of S, then S \ T ⇢ Te (recall Definition 5.6). Therefore, it suffices to prove an upper bound for the number of B3 -sets S satisfying S \ T ⇢ Te. For that we shall apply Lemma 3.3 to the graph CT [Te]. Therefore we have to show that CT [Te] satisfies the conditions of the lemma. We need the following claim. Claim 5.12. The set Ted1/"e is empty.

20

463 464 465 466 467 468

Proof. Recall that Ted1/"e is the set of all x such that T [{x} is a B3 -set and there is no graph Gd1/"e ⇢ T [{x} satisfying both conditions (a) and (b) of Definition 3.9 with i = d1/"e and T [ {x} in place 2 of T . The claim will follow after we show that for any x such that T [{x} is a B3 -set, if we let Gd1/"e be the complete graph K(T [ {x}) on T [ {x}, then conditions (a) and (b) hold. Condition (a) follows immediately for Gd1/"e = K(T [ {x}). As for (b), observe first that, as a little thought shows, we have RGd1/"e (z)  |T | + 1 for all z since T [ {x} is a B3 -set. Hence QGd1/"e  ne|T |+1  en exp

469

470

471 472

475 476

477

478

479

480 481

X 1◆ , j

|T |+1

(93)

j=1

⇤

The following claim is a simple consequence of Corollary 5.11 and Claim 5.12. Claim 5.13. Suppose that T is a ( , ")-bounded set with cardinality at least n1/100 and less than s1 6" . For any set A ⇢ Te with d1/"es

we have eCT (A) = ⌦

474

nd1/"e"

which establishes (b) and proves the claim.

|A|

473

✓

2+8"

✓ ↵3 d1/"e

n,

1 |A|

(94)

2

|T | n" (log n)3

◆

.

(95)

Proof. Let A ⇢ Te be as in the statement of the claim. By Claim 5.12 and the pigeonhole principle, there must be some i 2 {0, 1, . . . , d1/"e 1} such that |A \ Tei | |A|/d1/"e s 2+8" n. Applying Corollary 5.11 to A \ Tei in place of A yields that ✓ 3 ◆ ↵i |A \ Tei |2 eCT (A) = ⌦ . (96) |T | n" (log n)3 Since ↵i

↵d1/"e

1

and |A \ Tei | = ⇥(|A|), the claim follows.

⇤

We are finally ready to prove Proposition 3.11.

Proof of Proposition 3.11. In view of Claim 5.13, the graph CT [Te] satisfies (16) for ✓ ↵3 ◆ d1/"e 1 =⌦ and R = n = d1/"es 2+8" n 1. |T | n" (log n)3

(97)

Set q = (log n)/ and note that (18) is satisfied as well. Moreover, since the assumptions of Proposition 3.11 require that s > n1/5 and |T |  s1 6" , we have ✓ ◆ ⇣ ⌘ 4 d1/"e +1 (24) " (log n) q = O |T | n 3 = O s1 6" s5" (log n)32 = o(s). (98) ↵d1/"e 1 21

482 483

484 485

From Lemma 3.3 we conclude that the number of extensions of T into a B3 -set of size s such that T is a maximal ( , ")-bounded subset is at most ✓ ◆✓ ◆ ✓ ◆s q |T | n d1/"es 2+8" n d1/"ee s |T | n q s q |T | s2 8" (s q |T |) (99) ✓ ◆s 1 s |T | n . s3 8" o(1) In view of Lemma 4.1, a maximum ( , ")-bounded subset of a B3 -set of cardinality s always contains at least s1/7 elements, hence we can assume, without loss of generality, that n1/100 ⌧ s1/7  t = |T | < s1

486 487

t=s

✓ ◆ ✓ n s t n t s3 1/7

491 492 493 494

495 496 497 498 499

500 501

503

505 506



✓

n s3

8" o(1)

◆s

. ⇤

Fix an integer i 0 and real numbers " > 0, 1, D 5000 and 2 (0, 1] satisfying 2 ↵i /100⇠. Suppose that a set T ⇢ [n] with at least n1/100 elements satisfies P ,",i . Moreover, suppose that A ⇢ [n] and B ⇢ [2n] are such that the graph H = HT (A, B) satisfies the two conditions from the statement of the lemma. The fact that T satisfies P ,",i means that, in particular, we may choose a graph Gi on the vertex set T that satisfies (a) and (b) of Definition 3.9. Definition 6.1 (Special paths). A 4-path (a, b, a0 , b0 , a00 ) in H is said to be a Gi -special path, or simply a special path, if (a) a, a0 , a00 2 A and b, b0 2 B, (b) {b a, b0 a0 } and {b a0 , b0 a00 } are edges of Gi , and (c) the di↵erences b a, b0 a0 , b a0 , and b0 a00 are all distinct. Note that a 4-path (a, b, a0 , b0 , a00 ) between a and a00 2 A is special if, letting z1 = b a, z2 = b0 a0 , z3 = b a0 , and z4 = b0 a00 , we have a = (z1 + z2 )

(z3 + z4 ) and the zi s are all distinct.

(100)

We claim that for any a, a00 2 A, the number of special paths from a to a00 is at most 4RGi (a00 a). Indeed, if an ordered 4-tuple (z1 , z2 , z3 , z4 ) is a solution to (100), then the sequence of elements a, b := a + z1 , a0 := a + z1

504

8" o(1)

◆s

6. Proof of Lemma 4.5

(z1 z2 , z3 z4 ) 2 G2i , a00 502

1

This completes the proof of Proposition 3.11.

489 490

.

In particular, considering all possible choices of seed set T , the number of B3 -sets that do not contain any ( , ")-bounded subset of size larger than s1 6" is at most 1 6" sX

488

6"

z3 , b0 := a + z1

z3 + z2 , a00 = a + z1

z3 + z2

z4

(101)

forms a special path in H provided that a0 2 A and b, b0 2 B. Any solution to (100) remains a solution after swapping z1 with z2 or z3 with z4 . Therefore, it follows from the definition of RGi (see (20)) that the number of solutions to (100) is exactly 4RGi (a00 a). (For completeness, 22

a

b

a0

a + b0

a00

contained in NGi (b

a) + a0

b

b0

Figure 4. Counting semi-special paths extending P = (a, b, a0 ) in the graph H 0 from Claim 6.3. The first two edges are determined by P and the third edge {a0 , b0 } must be such that {b a, b0 a0 } 2 Gi . In view of the properties of H 0 , most of the H 0 -neighbors of a0 produce extensions of P to a semi-special path. Note that the fourth edge may be any edge incident to b0 except for {a0 , b0 } and possibly {b0 , a + b0 b}, {b0 , a0 + b0 b} and {b0 , a}. For instance, if a + b0 b is a neighbor of b0 , then it cannot be used to produce a semi-special path since the di↵erence b0 (a + b0 b) = b a would repeat the di↵erence of the first edge {a, b}. 507 508

509

we remark that not all solutions need to define paths in H since A and B are just subsets of [n] and [2n].) We conclude that the total number N of special paths in H satisfies N = O |CeGi |

(see Definition 3.5). Recalling (23), given that Gi satisfies (b) of Definition 3.9, we have 4RGi (a

510 511

512

(102)

00

a)  4 log QGi  4 n

i"

|T | X 1 j=1

j

 4 ni" log n.

(103)

In view of Proposition 3.6, inequalities (102) and (103) tell us that Lemma 4.5 will be proved if we establish the following claim. Claim 6.2. The total number N of special paths satisfies ✓ 2 ◆ |A| D2 |T |2 N =⌦ . (log n)2

(104)

514

In order to prove Claim 6.2, we will first construct a subgraph H 0 ⇢ H satisfying certain properties that will enable us to estimate the number of special paths N in H.

515

Claim 6.3. There exists d

513

D/16 and H 0 ⇢ H with vertex classes A0 ⇢ A and B 0 ⇢ B such that

degGi (b a) 1 4↵i / |T | for every (a, b) 2 H 0 ; 0 |H | |H|/8 log n; degH 0 (a) |T |/16 log n for every a 2 A0 ; d  degH 0 (b)  12d for every b 2 B 0 .

519

(i ) (ii ) (iii ) (iv )

520

We postpone the proof of Claim 6.3 and now establish Claim 6.2.

516 517 518

521 522

Proof of Claim 6.2. Let P = (a, b, a0 ) be an arbitrary path of length two in the graph H 0 obtained from Claim 6.3, with a, a0 2 A0 and b 2 B 0 . Consider all possible extensions of this path to a path 23

523

of length four, say (a, b, a0 , b0 , a00 ) with the condition that the di↵erences b

524 525 526

527 528 529 530 531

a0 , b0

a0 , b0

534 535 536 537 538

degH 0 (a0 ) . 150 Consequently, at least 99.3% of the neighbors b0 of a0 in H 0 are such that {b (24)

|T |  400⇠ |T | 

540

b0 2X

544 545

(108)

b 2X

(110)

b0 2X c

Since the degrees in B 0 are all in [d, 12d], we get X 12 12 X degH 0 (b0 )  12d |X c |  d |X|  degH 0 (b0 ). 99 99 0 c 0

(111)

b 2X

Hence, the total number of 4-paths starting with P = (a, b, a0 ) is bounded from above by ✓ ◆ ✓ ◆ (109) 12 X 4 12 100 0 1+ degH 0 (b )  1+ NP < NP . 99 0 99 98 3 b 2X

543

a0 } 2 Gi . Let

where in the last inequality we used the fact that d D/16 > 200. On the other hand, the total number of 4-paths starting with P is at most X X X degH 0 (b0 ) = degH 0 (b0 ) + degH 0 (b0 ).

b 2X

542

a, b0

a 0 } 2 Gi

b 2X

b0 2NH 0 (a0 )

541

a, b0

(107)

and X c = NH 0 (a0 ) \ X. Note that |X| 0.993 |NH 0 (a0 )| 1 0.99|NH 0 (a0 )|. For each b0 2 X, we have at least degH 0 (b0 ) 4 d 4 possible choices for a00 2 NH 0 (b0 ) that produce a semi-special path, namely, the only requirement is that b0 a00 must be di↵erent from the other three di↵erences and a00 cannot coincide with a (in fact, one sees that this last condition is automatically satisfied, if one recalls that T is a B3 -set). See Figure 4 for an illustration. From the discussion above, the number NP of semi-special paths that start with P satisfies ✓ ◆ X X 4 X 0 NP (degH 0 (b ) 4) 1 degH 0 (b0 ) 0.98 degH 0 (b0 ), (109) d 0 0 0 b 2X

539

(105)

and the di↵erences b a, b a0 , b0 a0 , b0 a00 are all distinct. This means that the paths P ! and P are in fact special (recall Definition 6.1). We shall later use this simple fact. Since H 0 satisfies (iii ), we have degH 0 (a0 ) |T |/16 log n. Moreover, by condition (i ), we have degGi (b a) 1 4↵i / |T |. As we require that 2 ↵i /100⇠, it follows that the number of non-neighbors of b a in Gi is at most

X = b0 2 NH 0 (a0 ) \ {b} : {b 533

a00

are all distinct and, moreover {b a, b0 a0 } 2 Gi . Call such (oriented) paths semi-special. Note that if both P ! = (a, b, a0 , b0 , a00 ) and P = (a00 , b0 , a0 , b, a) are semi-special, then we must have both {b a, b0 a0 }, {b0 a00 , b a0 } 2 Gi , (106)

4↵i

532

a, b

(112)

Let N4 be the total number of paths in H 0 of length 4 starting and ending in A0 . We proved above that the number NP of semi-special paths that start with P corresponds to more than 3/4 of the total number of 4-paths that starting with P . Since our argument holds for every P , we 24

546 547 548 549 550

551

552

553 554 555 556 557 558 559 560

561

conclude that there are more than 3N4 /4 semi-special paths in H 0 . Considering the involution that takes 4-paths P = (a, b, a0 , b0 , a00 ) to their reverse P = (a00 , b0 , a0 , b, a), we see that more than 1/2 of the 4-paths in H 0 starting and ending in A0 are semi-special in both directions, and thus are special. That is, there are more than N4 /2 special paths in H 0 . Finally, we estimate N4 by first picking an edge ab 2 H 0 , then picking a neighbor a0 6= a of b, and so on. This yields ✓ ◆ |H| |T | N4 (d 1) 1 (d 2), (113) 8 log n 16 log n

It only remains to prove Claim 6.3. Proof of Claim 6.3. This proof will be divided into three simple steps. First we define a set L of low degree vertices in Gi and show that this set is quite small. In the second step, we partition the class B according to the degrees of the vertices in HT \L (A, B) and select one part Bj that is incident to a good fraction of the edges while at the same time Bj has vertices with roughly the same degree. Finally, we delete the vertices of low degree in HT \L (A, Bj ) to obtain the desired graph. We assume that n, and therefore |T |, are sufficiently large for the calculations that follow to hold. Let L = x 2 T : degGi (x) < 1 4↵i / ) |T | . (114) Note that 2 e(Gi ) =

X

x2T 562

degGi (x)  |L| 1

|L|)|T | = |T |2

4↵i / |T | + (|T |

4↵i

|T | |L|.

1

↵i |T |(|T |

1)

|T |2

2↵i |T |2 .

565 566

567 568

570

(117)

and hence |L||A|  |H|/2. Let H ⇤ = HT \L (A, B) ⇢ H be the subgraph of H consisting of all the edges ab 2 H (a 2 A, b 2 B) such that b a 2 T \ L. It follows from (117) and the assumption of the lemma that e(H ⇤ ) = e(H) |L| |A| e(H)/2. (118) Since the graph H 0 that we construct in what follows is a subgraph of H ⇤ , it will satisfy (i ). Let I0 = [0, D/4), and Ij = [(D/4)ej 1 , (D/4)ej ) for j 1. For j 0, let Bj = {b 2 B : degH ⇤ (b) 2 Ij }.

569

(116)

A straightforward comparison of the two inequalities above yields the following (non-tight) bound |H| |L|  |T |  , 2 2|A|

564

(115)

On the other hand, it follows from the assumption on Gi (see Definition 3.9(a)) that 2 e(Gi )

563

⇤

whence the claim follows.

(119)

Note that Bj = ; for j log |T | since the maximum degree is at most |T |. Moreover, the number of edges incident to B0 is at most |B| D/4 = e(H)/4. In particular, by the pigeonhole principle, 25

571

there exists 1  j  log |T | such that there are at least e(H ⇤ ) e(H)/4 log |T |

572 573 574

575 576 577 578

580

(120)

edges of H ⇤ incident to Bj . b = H ⇤ [A [ Bj ]. Since we assume that H satisfies (I), it follows Set d = (D/16)ej 1 , and H from (120) that |T | |A| b e(H) . (121) 4 log n b is at least |T |/4 log n and the degrees In particular, the average degree of vertices of A in H

of vertices in Bj are all in [4d, 12d]. While there exists a vertex from A with degree smaller b than 16 log n |T |, or a vertex from B with degree less than d, remove the vertex from the graph H, together with all the incident edges. The number of edges deleted by this procedure is at most |A|

579

e(H) 4 log n

1 b |T | + |B| d  e(H). 16 log n 2

(122)

b Hence, at least e(H)/2 e(H)/8 log n edges remain after the deletion procedure above. Let H 0 be the graph obtained after the procedure and observe that it satisfies (ii ), (iii ), and (iv ). ⇤

581

7. Concluding remarks

582

In this whole paper, we considered to be an arbitrary, but fixed positive real number. Our argument allows one to take some function = (n) with ! 0 as n ! 1. Here, we opted for simplicity and did not attempt to optimize the argument to obtain the smallest possible = (n). We close by restating our conjectured answer (see [4]) to the problem addressed in Section 2. We believe that Theorem 2.1, concerning the cardinality of the largest B3 -sets contained in the random sets [n]m , is a particular case of a more general result.

583 584 585 586 587

588 589 590

Conjecture 7.1. Let h 2 be an integer. Suppose 0  a  1 is a fixed constant and m = m(n) = (1 + o(1))na . Then, asymptotically almost surely, we have Fh ([n]m ) = nb+o(1) , where b = b(a) is given by 8 > for 0  a  1/(2h 1), >
> > :

1/(2h a/h

1)

for 1/(2h

1)  a  h/(2h

for h/(2h

1),

(123)

1)  a  1.

594

The fact that b(a) is at least as large as stated in (123) is proved in [4]. On the other hand, a routine argument shows that, if true, Conjecture 1.3 implies the upper bound for b(a) conjectured in (123). The case h = 2 of Conjecture 7.1 is proved in [10, 11] and we established the case h = 3 in this paper.

595

References

591 592 593

596 597

[1] R. C. Bose and S. Chowla, Theorems in the additive theory of numbers, Comment. Math. Helv. 37 (1962/1963), 141–147. 26

598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622

623 624 625 626 627 628 629 630 631

632 633 634

[2] P. J. Cameron and P. Erd˝ os, On the number of sets of integers with various properties, Number theory (Ban↵, AB, 1988), de Gruyter, Berlin, 1990, pp. 61–79. [3] S. Chowla, Solution of a problem of Erd˝ os and Tur´ an in additive-number theory, Proc. Nat. Acad. Sci. India. Sect. A. 14 (1944), 1–2. [4] D. Dellamonica, Jr., Y. Kohayakawa, S. J. Lee, V. R¨ odl, and W. Samotij, On the number of Bh -sets, submitted. [5] P. Erd˝ os, On a problem of Sidon in additive number theory and on some related problems. Addendum, J. London Math. Soc. 19 (1944), 208. [6] P. Erd˝ os and P. Tur´ an, On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc. 16 (1941), 212–215. [7] B. Green, The number of squares and Bh [g] sets, Acta Arith. 100 (2001), no. 4, 365–390. [8] H. Halberstam and K. F. Roth, Sequences, second ed., Springer-Verlag, New York, 1983. [9] D. J. Kleitman and K. J. Winston, On the number of graphs without 4-cycles, Discrete Math. 41 (1982), no. 2, 167–172. [10] Y. Kohayakawa, S. Lee, and V. R¨ odl, The maximum size of a Sidon set contained in a sparse random set of integers, Proceedings of the 22nd Annual ACM–SIAM Symposium on Discrete Algorithms (SODA 2011) (Philadelphia, PA, USA), Society for Industrial and Applied Mathematics, 2011, pp. 159–171. [11] Y. Kohayakawa, S. J. Lee, V. R¨ odl, and W. Samotij, The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers, Random Structures Algorithms, to appear. [12] K. O’Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin. (2004), Dynamic surveys 11, 39 pp. (electronic). [13] D. Saxton and A. Thomason, Hypergraph containers, arXiv:1204.6595 (2012). [14] S. Sidon, Ein Satz u ¨ber trigonometrische Polynome und seine Anwendung in der Theorie der Fourier-Reihen, Math. Ann. 106 (1932), no. 1, 536–539. [15] J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), no. 3, 377–385. Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA (D. Dellamonica Jr., Y. Kohayakawa and V. R¨ odl) E-mail address: [email protected], [email protected] ´ tica e Estat´ıstica, Universidade de Sa ˜ o Paulo, Rua do Mata ˜ o 1010, 05508– Instituto de Matema ˜ 090 Sao Paulo, Brazil (Y. Kohayakawa) E-mail address: [email protected] Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea (S. J. Lee) E-mail address: [email protected]

School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, and Trinity College, Cambridge CB2 1TQ, UK (W. Samotij) E-mail address: [email protected]

27