The final size of the C4-free process Michael E. Picollelli∗ August 7, 2011

Abstract We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of C4 . We show that, with probability tending to 1 as n → ∞, the final graph produced by this process has maximum degree O((n log n)1/3 ) and consequently size O(n4/3 (log n)1/3 ), which are sharp up to constants. This confirms conjectures of Bohman and Keevash and of Osthus and Taraz, and improves upon previous bounds due to Bollob´as and Riordan and Osthus and Taraz.

1

Introduction

The H-free process, where H is a fixed graph, is the random graph process which begins with a graph G(0) on n isolated vertices. The graph G(i) is then formed by adding an edge ei selected uniformly at random from the pairs which neither form edges of G(i − 1) nor create a copy of H in G(i − 1) + ei . The process terminates with a maximal H-free graph GH with a random number MH of edges. Note that throughout this paper, H-free means ‘does not contain H as a (not necessarily induced) subgraph’. Erd˝os, Suen and Winkler [6] suggested this process as a natural probability distribution on maximal H-free graphs, and asked for the typical properties of GH , such as size and independence number. They considered the odd-cycle-free and triangle-free processes, establishing that the former terminates with Θ(n2 ) edges with high probability1 , and that, for√some positive constants c1 , c2 , c3 , whp c1 n3/2 ≤ MK3 ≤ c2 n3/2 log n and were improved by Spencer [13], who further conjecα(GK3 ) ≤ c3 n log n. These bounds √ 3/2 tured that whp MK3 = Θ(n log n). (We mention that the earliest result on an H-free process is due to Ruci´ nski and Wormald [11], who showed that the maximum-degree d process terminates in a graph with bnd/2c edges with high probability - here H is the star graph K1,d+1 .) ∗

Department of Electrical & Computer Engineering, University of Delaware, Newark, DE, USA. E-mail: [email protected] 1 We say a sequence of events An occurs with high probability, or simply whp, if limn→∞ P[An ] = 1.

1

More general H-free processes, where H satisfies an additional density condition, were first studied by Bollob´as and Riordan [5] and by Osthus and Taraz [9] independently. We say a graph H is 2-balanced if e(H) ≥ 3, v(H) ≥ 3, and e(H) − 1 e(F ) − 1 ≥ v(H) − 2 v(F ) − 2 for all proper subgraphs F of H with v(F ) ≥ 3, and strictly 2-balanced if the inequality is sharp for all such F ; examples of such graphs include cycles, complete graphs, and complete bipartite graphs Kr,r , r ≥ 2. For this class of graphs, Bollob´as and Riordan established general lower bounds on MH , and upper bounds for H ∈ {C4 , K4 } that match to within a logarithmic factor. Osthus and Taraz then gave upper bounds for all strictly 2-balanced H that match to within a logarithmic factor. For H = C4 , the results of Bollob´as and Riordan yield MC4 = Ω(n4/3 ) and MC4 = O(n4/3 (log n)3 ) whp; Osthus and Taraz’s results improve the upper bound to MC4 = O(n4/3 log n) whp, and they further conjectured that whp the average degree of GC` is O((n log n)1/(`−1) ) for all ` ≥ 3. Evidence that the lower bound was not sharp came from Wolfovitz [16] who improved the lower bound on E (MH ) for regular strictly 2-balanced H by a factor of (log log n)1/(e(H)−1) . Finally, through an application of the differential equations method (for the general √ method and examples, see [19]), Bohman [2] showed MK3 = Θ(n3/2 log n) whp, confirming Spencer’s conjecture, and produced an improvement on the lower bound for MK4 . Subsequent work by Bohman and Keevash [3] established new lower bounds on MH for all strictly 2balanced H by producing lower bounds on the minimum degree of GH that hold whp, and they conjectured that the likely maximum degree of GH is at most a constant multiple of their lower bound. For the C4 -free process, their bound on the minimum degree is Ω((n log n)1/3 ), yielding MC4 = Ω(n4/3 (log n)1/3 ) whp. We mention some motivation for studying the H-free process comes in part from its connection to two classical areas of extremal combinatorics, Ramsey theory and Tur´an theory. Bounds on the independence number of GK3 found in [6] and [13] led to the best lower bounds on R(3, t) known at the time, and Bohman’s analysis [2] produced an improvement that matched Kim’s celebrated lower bound [8]. The analysis in [2] and [3] has also led to the best current lower bounds for the Ramsey numbers R(s, t), with s ≥ 4 fixed and t large, and the cycle-complete Ramsey numbers R(C` , Kt ) for ` ≥ 4 fixed and t large. The results for the Kr,r -free process in [16] and [3] have resulted in improvements on the best known lower bounds for the Tur´an numbers ex(n, Kr,r ) for r ≥ 5. However, the process has also become a subject of recent interest on its own, in part for aspects of Bohman and Keevash’s analysis that suggest the graph G(i) produced by the process resembles the random graph G(n, i), chosen uniformly at random from all i-edge graphs on n vertices, with the exception that it contains no copies of H. To establish their lower bound, they show that a wide range of subgraph extension variables, including the number of copies of a given H-free graph F , take roughly the same values in G(i) as in G(n, i), for i up to a small multiple of n2−(v(H)−2)/(e(H)−1) (log n)1/(e(H)−1) . (Similar results on subgraph counts in the K3 -free process were obtained by Wolfovitz [18].) In fact, the lower bound and conjectured upper bounds on MH in [3] correspond (within constant factors) to 2

the threshold for the random graph G(n, i) to have the property that the addition of any new edge creates a new copy of H, provided H is strictly 2-balanced (see [12]). It is also known ([7] and [14]) that sufficiently dense subgraphs are unlikely to appear in GH . Very recently, Warnke [15] and Wolfovitz [17] have independently given upper bounds on MK4 that match Bohman’s lower bound to within a constant factor. The author [10] has also established matching bounds for the case where H is the diamond graph, formed by removing an edge from K4 . (The diamond graph is 2-balanced but not strictly so.) Along with K3 , these are the only 2-balanced graphs containing a cycle for which such bounds on MH are currently known. Our aim is to add C4 to this list through the next result. Theorem 1. There exists κ > 0 such that ∆(GC4 ) ≤ κ(n log n)1/3 with high probability. This confirms the mentioned conjectures of Osthus and Taraz and of Bohman and Keevash for the C4 -free process. Combined with the lower bound given in [3], this has the following immediate corollary. Corollary 2. With high probability, MC4 = Θ(n4/3 (log n)1/3 ). From an upper bound established in [3], as well as known bounds on the independence number of C4 -free graphs with bounded maximum degree, we arrive at the next result easily. Corollary 3. With high probability, α(GC4 ) = Θ((n log n)2/3 ). An immediate consequence of this second corollary is that a typical graph produced by the C4 -free process will not essentially improve the lower bound on R(C4 , Kt ) given in [3].

1.1

Overview of the argument

The key to our argument is the following simple observation: suppose we fix a vertex v and a step i ≤ MC4 . If x and y are neighbours of v in GC4 but are not adjacent to v in G(i), then x and y have no common neighbours in G(i). Consequently, we can produce a bound of the form ∆(GC4 ) ≤ ∆(G(i)) + k by showing that every set K of k vertices contains two which have a common neighbour in G(i). To do so, we follow the methods of [2] and [3] and use the differential equations method to track a collection of random variables, defined in Section 2.3 below, which count extensions (in the sense of [12]) from pairs of vertices u, v ∈ K to vertices w, such that one of the pairs uw, vw forms an edge in G(i) while the other can still be selected as the next edge ei+1 . Our bounds will then imply that the number of such selectable pairs is sufficiently large that, whp, at least one is chosen by the process for every such K by some step i. An analogous although more sophisticated argument was used by Warnke to establish his bound on MK4 , and we mention that we have based much of the presentation of this paper on [15]. Such an approach appears suitable for bounding MH for more general choices of H, but significant technical complications can quickly arise. In particular, for many natural choices of random variables to track, the addition of a single edge in the process can result in drastic one-step changes which exceed the boundedness requirements of the differential equations 3

method, or in certain ‘pathological’ conditions (see the discussion preceding Lemma 8.3 in [3]) that significantly alter the expected one-step change (trend hypothesis) calculations. Avoiding these effects can require considerable care, such as in [3] (Section 11 particularly) and [15]. By our key observation, we need only track our random variables for a set K until the desired common neighbour is created, after which we may ‘ignore’ K. As a result, each vertex will have at most one neighbour in any given ‘unignored’ set K in G(i), which lets us avoid the technical complications for a fairly simple choice of random variables! This is also the primary reason our argument is so short, compared with [15] or [17]. However, we admit that no such simplification is apparent for most other choices of H, specifically H = Kr for r ≥ 4 or H = C` for ` ≥ 5. The remainder of this paper is organized as follows: in Section 2, we discuss the C4 -free process specifically, including relevant results from [3]. In Section 2.3, we define our random variables and state our main technical lemma, Lemma 6, followed by the proofs of Theorem 1 and Corollary 3. The remainder of the paper is then devoted to the proof of Lemma 6: in Section 3 we present a lemma from [3] which forms the basis for our differential equations method application, and we cover some necessary inequalities for our arguments in Section 3.1. Finally, we prove Lemma 6 in Section 4.

2 2.1

The C4-free process Definitions and notation

We let [n] = {1, . . . , n} be the vertex set of the process, and
let G(i) denote the graph given by the first i edges selected. G(i) naturally partitions [n] into three sets, E(i), O(i), and 2 C(i). E(i) is simply the edge set of the process. For a pair uv ∈ / E(i), we say uv is open, and uv ∈ O(i), if the graph G(i) + uv is C4 -free, and we define Q(i) = |O(i)|. Otherwise, we say uv is closed and uv ∈ C(i). For v ∈ [n], we let Ni (v) and di (v) denote the neighbourhood and degree, respectively, of v in G(i). For i ≥ 0 and for each pair of vertices uv ∈ O(i) ∪ C(i), we define Cuv (i) to be the set of pairs wz ∈ O(i) such that G(i) + uv + wz contains a copy of C4 which uses both uv and wz as edges. We mention that Cuv is defined in [3] as the set of ordered pairs; as in [2] or [15], we will work exclusively with unordered pairs. We introduce a continuous time variable t, and relate it to the process by setting t = t(i) = i/n4/3 . We fix constants µ, ε, V, W , which satisfy 0<µε

1 1 1   . W V 4

(The notation 0 < a  b means there is an increasing function f (x) so the arguments which follow are valid for 0 < a < f (b).) Given these constants, we define p = n−2/3 ,

m = µ(log n)1/3 · n4/3 ,

4

and

tmax = µ(log n)1/3 .

(1)

We further define functions q(t), c(t), P (t), e(t) as well as parameters s = s(n) and se = se (n) as follows: q(t) = exp(−8t3 ),

c(t) = 24t2 exp(−8t3 ),

3

P (t)

P (t) = W (t + t), 2

4/3

s(n) = n p = n

e(t) = e ,

− 1,

1/8−ε

and

se (n) = n

.

(2) (3) (4)

We assume that ε and µ are chosen sufficiently small that e(t) and q(t)−1 are at most nε for 0 ≤ t ≤ tmax , and se = n1/8−ε  nε , so e(t)/se = o(1) uniformly in 0 ≤ t ≤ tmax . We will present additional bounds on µ, ε, V and W sufficient for our argument in Section 3.1. We use the notation “±” in two distinct ways throughout this paper. The notation a ± b will be taken to mean the interval {a + xb : −1 ≤ x ≤ 1}; distinct instances of ± used this way in the same expression will be treated independently, i.e. (a ± b)(c ± d) will be taken to mean {(a + x1 b)(c + x2 d) : −1 ≤ x1 , x2 ≤ 1}. We will also write a = b ± c instead of a ∈ b ± c. For a sequence of random variables A(1), A(2), . . ., we will use A± to denote pairs of sequences of nonnegative random variables A+ (1), A+ (2), . . . and A− (1), A− (2), . . . such that A(i + 1) − A(i) = A+ (i) − A− (i). Similarly, for a differentiable function f (t), we will use f ± to denote pairs of nonnegative functions f + , f − satisfying f 0 (t) = f + (t) − f − (t), noting that this is a nonstandard use of the notation f + , f − .

2.2

The lower bound - results of Bohman and Keevash

Bohman and Keevash [3] established their lower bound on the H-free process by showing that certain random variables are tightly concentrated throughout the initial m steps. As we do not require the full strength of their results, we summarize the relevant consequences for the C4 -free process in the following theorems. Theorem 4 (Bohman and Keevash, [3]). Let Ti∗ denote the event that the following hold for 0 ≤ i ≤ i∗ : 1.

   1 n2 e(t) q(t) ± ; Q(i) = 1 ± se se 2

(5)

2. for all v ∈ [n],  di (v) =

e(t) 1± se

  1 2t ± np, se

(6)

and so ∆(G(i)) ≤ 4tmax np; 3. for all uv ∈ O(i) ∪ C(i),    e(t) 12 p−1 2 |Cuv (i)| = 1 ± 24t q(t) ± , se se 2 5

(7)

and for all distinct uv, u0 v 0 ∈ O(i), |Cuv (i) ∩ Cu0 v0 (i)| ≤ n−1/4 p−1 ; and

(8)

4. for every pair of distinct vertices u and v, there are at most n12ε paths of length 3 between them in G(i). Then Tm holds with high probability. Theorem 5 (Bohman and Keevash, [3]). With high probability, α(G(m)) ≤ 3µ−1 (n log n)2/3 . Recalling that we may choose ε and µ so that e(t)/se = o(1) and q(t) ≥ n−ε  1/se , Tm implies Q(m) > 0 and consequently the lower bound MC4 ≥ µn4/3 (log n)1/3 holds whp. Equation (5) follows immediately from Theorem 1.4 of [3]. Equation (6) follows similarly, while the bound on ∆(G(i)) follows as e(t)/se ≤ 1/3 and 1/se ≤ tmax . Equations (7) and (8) follow from Corollary 6.2 and Lemma 8.4 of [3], respectively. We mention that the phrasing of Lemma 8.4 suggests that uv and u0 v 0 are fixed. However, as Lemma 8.4 is shown to be a consequence of a constant (depending on H) number of applications of Lemma 5.2, which has exponentially small failure probability (conditioned on their event Gm ), (8) follows for all such pairs and steps i, 0 ≤ i ≤ m, with high probability. 1 , which we will The bound given in 4. follows from Lemma 5.1 in [3], provided ε < 24 3 4−2 4·3·ε 12ε assume. In particular, Lemma 5.1 yields the bound (p n ) · n = n on the number of paths of length 3 between any fixed pair of vertices in G(m), and hence in G(i) for 0 ≤ i ≤ m.
Furthermore, Lemma 5.1 has an exponentially small failure probability, covering our n2 applications. Finally, Theorem 5 follows from the proof of Theorem 1.9 of [3] (specifically, Lemmas 11.3 and 12.1).

2.3

The upper bound - proofs of Theorem 1 and Corollary 3

Let β > 0 be a fixed constant satisfying β>

4 , µ2

(9)

and define k = β · (n log n)1/3 .

(10)

We say a k-set K is covered in G(i) if a common neighbour exists in G(i) for some pair of vertices in K; K is uncovered otherwise. We recall from the discussion in Section 1.1 that if every k-set is covered in G(m), then for all v ∈ [n], dMC4 (v) ≤ dm (v) + k = O((n log n)1/3 ), the desired bound.
If a set K is covered in G(i), then there exist vertices u, v, w such that uv ∈ K2 and uw, vw ∈ E(i). We note that the order of u and v is not essential, and we expect that it is likely (but not necessary!) that the common neighbour w does not lie in K. We
therefore restrict our attention to certain subsets of K2 × ([n] \ K). We will write elements 6

of of

K × ([n] \ K) as (uv, w) but will refer to them as triples to avoid confusion with elements 2

[n] [n] . We will also identify each such triple (uv, w) with the subset {uw, vw} of . 2 2
[n] We introduce the following definitions: given K ∈ k and 0 ≤ i ≤ m, we define






   K XK (i) = (uv, w) ∈ × ([n] \ K) : uw, vw ∈ O(i) , and 2     K YK (i) = (uv, w) ∈ × ([n] \ K) : |{uw, vw} ∩ O(i)| = |{uw, vw} ∩ E(i)| = 1 . 2 We call triples in XK (i) open with respect to K, and triples in YK (i) partial with respect to K. We note that for a partial triple (uv, w) ∈ YK (i), where, say, uw ∈ O(i), if ei+1 = uw then K is covered in all steps i0 ≥ i + 1. More importantly, if K is uncovered in G(i), then |Ni (v) ∩ K| ≤ 1 for all v ∈ [n]; this immediately implies that each open pair in O(i) can lie in at most one partial triple in YK (i). Intuitively, the probability that a given pair of vertices xy is open at time t = t(i) is ≈ q(t), while the probability that xy is an edge is ≈ 2tp. It is then reasonable to suspect   k q(t)2 2 · k n, and |XK (i)| ≈ · (n − k) · q(t)2 ≈ 2 2   k |YK (i)| ≈ 2 (n − k)(q(t))(2tp) ≈ 2tq(t) · k 2 np. 2 The following lemma shows that these estimates are correct for uncovered K.
Lemma 6. With high probability, for all i, 0 ≤ i ≤ m, and K ∈ [n] , if K is uncovered in k G(i) then    1 q(t)2 e(t) ± 3ε k 2 n, and |XK (i)| = 1 ± 3ε n 2 n    e(t) 1 |YK (i)| = 1 ± 3ε 2tq(t) ± 3ε k 2 np. n n The proof of Lemma 6 is the primary technical challenge of this paper, and will be presented in Sections 3 and 4. We now turn our attention to proving Theorem 1. Proof of Theorem 1. We may assume the conclusions of Theorem 4 and Lemma 6 hold, as their failure probability is o(1). We also assume that µ, ε, V and W are chosen so that q(t)−1 and e(t) are at most nε on [0, tmax ], and se ≥ n3ε . By Theorem 4, ∆(G(m)) ≤ 4µ(n log n)1/3 . Letting κ = 4µ + β, we establish the bound in Theorem 1 by showing that whp every k-set is covered in G(m). Given an uncovered K at step i, as ei+1 is chosen uniformly at random from Q(i) open pairs, and as each partial triple (uv, w) ∈ YK (i) contains a unique open pair, the probability K (i)| . We bound the probability that K remains uncovered in G(i + 1) is at most 1 − |YQ(i) some k-set K remains uncovered for all steps i, m/2 ≤ i ≤ m. For n sufficiently large, 7

m/2 ≥ n4/3 , so in this range of i we may assume t = t(i) ≥ 1. Thus, if K is uncovered in G(i) with m/2 ≤ i ≤ m, then as se ≥ n3ε , e(t)/n3ε ≤ 1/3 and 1/n3ε ≤ q(t)/2 ≤ tq(t)/2 for tmax /2 ≤ t ≤ tmax , we have (1 ± e(t)/n3ε )(2tq(t) ± 1/n3ε )k 2 np 2/3 · (3/2)tq(t)k 2 np tmax k 2 p |YK (i)| = ≥ ≥ . Q(i) (1 ± e(t)/se )(q(t) ± 1/se )n2 /2 4/3 · (3/2)q(t)n2 /2 2n Thus, the probability that some k-set remains uncovered for m/2 ≤ i ≤ m is at most   m/2   n tmax k 2 p tmax k 2 pm k 1− ≤ n exp − k 2n 4n   µ(log n)1/3 (β(n log n)1/3 )2 · n−2/3 · µn4/3 (log n)1/3 k = n exp − 4n  2 2 1/3  4/3 µ β n (log n) = nk exp − . 4 As nk = exp(βn1/3 (log n)4/3 ), this is o(1) provided µ2 β 2 /4 > β, i.e. (9) holds. Corollary 2 is immediate, so we turn to the proof of Corollary 3. The upper bound on α(GC4 ) follows from Theorem 5, as α(GC4 ) ≤ α(G(m)). For the lower bound, we apply a lemma from [4] bounding the independence number of graphs with few triangles. (Similar bounds are known for a wider class of H-free graphs - see [1].) Lemma 7 ([4], Lemma 12.16 (ii)). Let G be a graph on n vertices with average degree at most d and at most h triangles. Then    h 1 1 n · log d − log . α(G) ≥ 10 d 2 n Proof of Corollary 3. By Theorem 1, with high probability the average degree of GC4 is at most κ ˆ = κ(n log n)1/3 , where κ > 0 is a fixed constant. As GC4 is C4 -free, each edge lies on at most one triangle, so GC4 has at most MC4 /3 triangles. Taking d = κ ˆ and h = κ ˆ n in Lemma 7, and observing that log κ ˆ ≥ (log n)/3 for n sufficiently large, we have   1 1 n 1 · log κ ˆ ≥ (n log n)2/3 . α(GC4 ) ≥ 10 κ ˆ 2 60κ

3

The differential equations method

We will prove Lemma 6 by appealing to an approach to the differential equations method presented in Lemma 7.3 of [3], which we present below (Lemma 8). In Section 3.1, we present necessary bounds on our constants and additional inequalities which will simplify some of the more technical calculations. The proof of Lemma 6 will then follow in Section 4. We reproduce from [3] the setup for this lemma: suppose we have a stochastic graph process G(0), G(1), . . . defined on [n], where n is large. Let r be a fixed positive integer, and for j ∈ [r], let kj , Sj be parameters (which can depend on n). 8


Suppose for each j ∈ [r] and A ∈ [n] , there is a sequence of random variables Xj,A (i), kj defined for i = 0, . . . , m and measurable with respect to the underlying graph process. We suppose further that + − Xj,A (i + 1) − Xj,A (i) = Xj,A (i) − Xj,A (i), + − where Xj,A (i), Xj,A (i) ≥ 0. We relate these sequences to functions on [0, ∞) by letting t = i/s for some function s = s(n) that tends to infinity. The goal is then to argue that, for some collection xj (t) of continuous functions,

Xj,A (i) ≈ xj (t)Sj
for all j ∈ [r], A ∈ [n] , and i = 0, . . . , m. We view 1 ≤ j ≤ r as the type of random variable, kj and the set A as giving its position in the graph. The parameter Sj is the size-scaling for the jth type of random variable. Lemma 8 ([3], Lemma 7.3). Let 0 < ε < 1 and c, C > 0 be constants, and suppose for each j ∈ [r] we have a parameter sj (n) and functions xj (t), ej (t), θj (t), γj (t) that are smooth and nonnegative for t ≥ 0. For i∗ = 1, 2, . . . , m, let Gi∗ be the event that    θj (t) ej (t) xj (t) ± Sj Xj,A (i) = 1 ± sj sj
. Suppose there is also a decreasing sequence for all 1 ≤ i ≤ i∗ , 1 ≤ j ≤ r, and A ∈ [n] kj 2 of events Hi , 1 ≤ i ≤ m, such that P[∃i ≤ m : Gi ∧ ¬Hi ] = o(1), and that the following conditions hold: 1. (Trend hypothesis) When conditioning on Gi ∧ Hi , we have  
hj (t) Sj ± ± E Xj,A = xj (t) ± , (11) 4sj s
, where x± for all j ∈ [r] and A ∈ [n] j (t) and hj (t) are smooth nonnegative functions kj such that − x0j (t) = x+ and hj (t) = (ej xj + γj )0 (t); j (t) − xj (t) 2. (Boundedness hypothesis) For each j ∈ [r], conditional on Gi ∧ Hi , we have ± Xj,A (i) <

Sj ; 2 sj kj nε

(12)

3. (Initial
conditions) For all j ∈ [r], we have γj (0) = 0 and Xj,A (0) = Sj xj (0) for all A ∈ [n] ; kj 2

That the proof of Lemma 7.3 of [3] utilizes this condition rather than the condition limn→∞ P[Hm | Gm ] = 1 stated in [3] was first pointed out by Warnke - see Appendix A.1 of [15].

9

4. We have n3ε < s < m < n2 , m ≤ nε/2 s, s ≥ 40Cs2j kj nε , n2ε ≤ sj < n−ε s, inf θj (t) + ej (t)xj (t)/2 − γj (t)/2 > c, t≥0 Z ∞ ± 0 sup |xj (t)| < C, sup |xj (t)| < C, |x00j (t)| dt < C, t≥0

t≥0

0

sup |hj (t)| < nε ,

Z

0≤t≤m/s

m/s

|h0j (t)| dt < nε .

0

Then P[Gm ∧ Hm ] → 1 as n → ∞. We mention that the assumption in 1. is that whenever Gi ∧ Hi holds, the conditional ± expectation of Xj,A given G(i) satisfies the bounds stated in (11). This will be implicit in our verification in Sections 4.1.1 and 4.2.1 below.

3.1

Additional inequalities and the constants µ, ε, V , W

As much of the remainder of this paper will be focused on verifying the conditions of Lemma 8, we take the opportunity now to gather a few simple inequalities to aid in the calculations. First, in addition to the constraints on µ, ε, V and W implicit in [3], the following bounds suffice for our application: V ≥ 14,

W ≥

224V e2V , log 2

ε≤

1 , 72

and µ is taken sufficiently small so that eP (t) ≤ nε/2 for t ∈ [0, tmax ], provided n is sufficiently large. From these bounds, (1)-(4), and (10), for all t in [0, tmax ] and n sufficiently large, we note that 1 ≤ q(t)−1 ≤ q 2 (t)eP (t) , se ≥ n8ε , and n1/3 ≤ k ≤ n1/3+ε . We also observe that for a fixed polynomial F (t) and any α > 0, as tmax = µ(log n)1/3 = no(1) , we may assume |F (t)| ≤ nα on [0, tmax ]. Finally, we present bounds on Q(i) and |Cuv (i)| which follow from Theorem 1. First, from (5) and max{q(t)−1 , e(t)} ≤ nε/2 , it follows that on the event Ti∗ , we have      1 n2 q(t) 2nε e(t) q(t) ± ⊆ 1± n2 Q(i) = 1 ± se se 2 2 se for 0 ≤ i ≤ i∗ , which we note implies Q(i) ≥ n2−ε , and consequently     1 2 3nε 1 2 n2ε 1 ⊆ . = 1± ± Q(i) q(t) s e n2 q(t) se n2 Similarly, for all uv ∈ O(i) ∪ C(i), as c(t) = 24t2 q(t) = o(nε/2 ), from (7) we have      e(t) 12 p−1 nε p−1 |Cuv (i)| = 1 ± c(t) ± ⊆ c(t) ± , se se 2 se 2 which yields the simple estimate |Cuv (i)| ≤ n2/3+ε . 10

(13)

(14)

4

Proof of Lemma 6

We turn now to proving Lemma 6 through an application of Lemma 8, and we assume, wherever necessary, that n is sufficiently large. We recall s = n4/3 , and we define, for t ≥ 0, x(t) = q(t)2 /2, y(t) = 2tq(t),

x+ (t) = 0, y + (t) = 2q(t),

x− (t) = c(t)q(t) = 24t2 q(t)2 , y − (t) = 2tc(t) = 48t3 q(t).

and

0 + − 0 + − so that x± , y ± are
nonnegative on [0, ∞), x = x − x and y = y − y . [n] For K ∈ k and 0 ≤ i ≤ m, let EK,i denote the event that K is uncovered in G(i) and Ti holds. As we are only interested in ensuring bounds that hold for uncovered K at each step i, and as Tm holds with high probability, it suffices to show the desired bounds hold on EK,i . Consequently, we will apply Lemma 8 to a modified collection of random variables that follow the correct trajectory deterministically on the event ¬EK,i . We remark that a similar modification is implicit in the proof of Theorem 1.4 in [3]. Therefore, for i = 0, 1, . . . , m, let ( |XK (i + 1) \ XK (i)| if EK,i holds, + XK (i) = x+ (t) · k 2 n/s otherwise,

( |XK (i) \ XK (i + 1)| − XK (i) = x− (t) · k 2 n/s and

if EK,i holds, otherwise,

( 3 2 |XK (0)| + k +kn−k 2 b XK (i) = bK (i − 1) + X + (i) − X − (i) X K K

if i = 0, otherwise.

Similarly, let ( |YK (i + 1) \ YK (i)| YK+ (i) = y + (t) · k 2 np/s ( |YK (i) \ YK (i + 1)| YK− (i) = y − (t) · k 2 np/s and

if EK,i holds, otherwise, if EK,i holds, otherwise,

( |YK (0)| YbK (i) = YbK (i − 1) + YK+ (i) − YK− (i)

bK (i) = |XK (i)| + It follows that when EK,i holds, X

k3 +kn−k2 2

if i = 0, otherwise. ≈ |XK (i)| and YbK (i) = |YK (i)|.

To set up our application, we recall m = µ(log n)1/3 n4/3 , and we let c = 1/4 and take C > 0 to be a sufficiently large constant, which
we will discuss further in Section 4.3. We let k1 = k2 = k, x1 = x, x2 = y, and for K ∈ [n] , we let k bK (i), X1,K (i) = X

S1 = k 2 n,

X2,K (i) = YbK (i), 11

and S2 = k 2 np.

b ± = X ± and Yb ± = Y ± , we will write the latter for ease of reading. As X K K K K We define, for t ≥ 0, γ(t) =


1 1 − exp(−224e2V · t) 4

and

θ(t) =

1 + γ(t). 2

(This specific choice of γ(t) was inspired by an early draft of [15].) For j ∈ {1, 2} we define the remaining error parameters as sj = n3ε ,

ej (t) = e(t),

γj (t) = γ(t)

and

θj (t) = θ(t),

where e(t) is defined in (3). Finally, we take the event Hi = Gi for all i, which trivially satisfies the necessary conditions. It follows that for 0 ≤ i∗ ≤ m, on the event Gi∗ , we have      θ(t) e(t)x(t) + 1 e(t) 2 b x(t) ± k n ⊆ x(t) ± k 2 n and XK (i) = 1 ± s1 s1 s1      (15) e(t) θ(t) e(t)y(t) + 1 2 2 YbK (i) = 1 ± y(t) ± k np ⊆ y(t) ± k np s2 s2 s2
for all K ∈ [n] and 0 ≤ i ≤ i∗ , by noting that e(t)/sj = o(1) and θ(t) ≤ 3/4. As k (k 3 + kn − k 2 )/2 = o(1) · k 2 n/s1 , the conclusions of Lemma 8 hold on the event Gm ∧ Tm , so it suffices to show Gm holds whp. We next note that the initial conditions are immediate: YbK (0) = 0 = y(0)k 2 np, and   2 3 2 bK (0) = k (n − k) + k + kn − k = k n = x(0)k 2 n. X 2 2 2 ± And, as intended, the trend and boundedness hypotheses follow deterministically for XK and ± YK on the event ¬EK,i . In particular, (11) is trivial, while (12) follows from the inequalities (which we will establish!) |x± | ≤ C, |y ± | ≤ C and s ≥ 40Cs2j kj nε . It remains, then, to show they hold on the event Gi ∧ EK,i .

4.1 4.1.1

Open triples Trend hypothesis

+ We note first that XK (i) = 0 for all i, implying (11) trivially, so we turn our attention to − XK . To simplify our calculations, all functions in the expressions which follow are assumed to be evaluated at t = t(i), and we will write q in place of q(t), etc.. To avoid potential confusion, we will use “e” to refer to the function defined in (3), and “e” to refer to the constant e = 2.718 . . .. − (i) if and only if Conditioned on EK,i , a triple (uv, w) ∈ XK (i) gets counted by XK ei+1 ∈ {uw, vw} ∪ Cuw (i) ∪ Cvw (i). As K is uncovered, u and v have no common neighbours

12

in G(i), and consequently uw ∈ / Cvw (i) and vice-versa. Therefore, by (8), (13) and (14), the probability (uv, w) is counted, conditioned on Gi ∧ EK,i , is       |Cuw (i) ∪ Cvw (i)| + 2 nε p−1 2 n2ε 1 −1/4 −1 = 2 c± ± (n p + 2) · ± Q(i) se 2 q se n2     2 n2ε p−1 2nε · ± ⊆ c± se q se n2   2c n3ε 1 ± , ⊆ q se s which follows as 2p ≤ n−1/4 ≤ s−1 and max{c(t), q(t)−1 } ≤ nε/2 . Summing this over all e (uv, w) ∈ XK (i) and using (15) yields    
ex + 1 2c n3ε 1 − 2 E XK (i) | Gi ∧ EK,i = x ± k n· ± s1 q se s    2 2xc 1 2c(ex + 1) (xs1 + ex + 1)n3ε k n ⊆ ± · + q s1 q se s   2
k n 1 , ⊆ x− ± · 48t2 (xe + 1) + 1 s1 s where the final containment follows as c(t) = 24t2 q(t), x(t) ≤ 1, se ≥ n7ε , and e(t) ≤ nε . It remains to show 48t2 (x(t)e(t) + 1) + 1 ≤ h1 (t)/4, where h1 (t) = (xe + γ)0 (t). From routine differentiation and the inequalities 2W ≥ 48 and γ 0 (t) ≥ 0, we have h1 (t) = −48t2 x(t)e(t) + W (3t2 + 1)x(t)eP (t) + γ 0 (t) ≥ W (t2 + 1)x(t)eP (t) . − As 1 ≤ 2x(t)eP (t) and W ≥ 572 = 4 · 144, (11) follows for XK by observing

48t2 (x(t)e(t) + 1) + 1 ≤ (144t2 + 2)x(t)eP (t) ≤ 4.1.2

W (t2 + 1)x(t)eP (t) . 4

Boundedness hypothesis

+ − , (12) follows as every open By the same observation as above, (12) is trivial for XK . For XK pair lies in at most k triples in XK (i), which, as 1/3 > 9ε, implies that on Gi ∧ EK,i , − |XK (i)| ≤ (|Cei+1 (i)| + 1) · k ≤ n2/3+2ε k <

4.2 4.2.1

nk S1 = . n7ε s21 knε

Partial triples Trend hypothesis

We first establish (11) for YK+ (i), conditioned on Gi ∧ Hi . A triple (uv, w) ∈ XK (i) enters YK (i + 1) if and only if ei+1 ∈ {uw, vw}, recalling that uw ∈ / Cvw (i) and vice-versa. Thus, 13

summing 2/Q(i) over the triples (uv, w) ∈ XK (i) and applying (13) and (15) yields    
ex + 1 4 2n2ε 1 + 2 E YK (i) | Gi ∧ EK,i = x ± k n· ± s1 q se n2   2  1 4ex 4 (xs1 + ex + 1)2n2ε k np 4x ± + + ⊆ q s1 q q se s    2 1 4 k np ⊆ y+ ± 2qe + + 1 , s2 q s recalling x(t) = q(t)2 /2 ≤ 1, s1 = s2 = n3ε , and se ≥ n7ε . As 1 ≤ q(t)−1 ≤ q(t)eP (t) , it suffices to show 7q(t)eP (t) < h2 (t)/4, where h2 (t) = (ye + γ)0 (t) = (2 − 48t3 )q(t)e(t) + (2tq(t))W (3t2 + 1)eP (t) + γ 0 (t) ≥ 2W (t3 + t)q(t)eP (t) + γ 0 (t).

(16)

If t < V /W < 1, then as t3 + t ≤ 2t, eP (t) ≤ eW (2V /W ) = e2V , and the bound follows as 2V

γ 0 (t) 224e2V · e−224e = 4 4

t

/4

≥

224e2V = 7e2V , 32

(17)

recalling W ≥ 224e2V · V /(log 2). Otherwise t ≥ V /W , and h2 (t)/4 is at least W (t3 + t) V · q(t)eP (t) ≥ · q(t)eP (t) ≥ 7q(t)eP (t) . 2 2

(18)

Next, we turn to YK− : for each triple (uv, w) ∈ YK (i), where, say, uw ∈ O(i), the probability that (uv, w) gets counted by YK− (i), conditioned on Gi ∧ EK,i , is        nε p−1 2 n2ε 1 c n3ε 1 |Cuw (i)| + 1 = c± +1 · ± ⊆ ± , Q(i) se 2 q s e n2 q se s following a similar calculation to that given at the start of Section 4.1.1. Applying (15),    
ey + 1 c n3ε 1 − 2 E YK (i) | Gi ∧ EK,i = y ± k np · ± s2 q se s   2  1 eyc c (ys2 + ey + 1)n3ε k np yc ⊆ ± + + q s2 q q se s   2
k np 1 ⊆ y− ± 48t3 qe + 24t2 + 1 , s2 s where the final containment follows as y(t) = 2tq(t) = o(nε/2 ). 14

As t2 ≤ t3 + t for t ≥ 0 and as W ≥ 288 = 4 · 72, we have 48t3 q(t)e(t) + 24t2 + 1 ≤ 72(t3 + t)q(t)eP (t) + 1 ≤

W 3 (t + t)q(t)eP (t) + 1. 4

By (16), to establish (11) it suffices to show 1≤

γ 0 (t) W 3 (t + t)q(t)eP (t) + , 4 4

which follows from (17) and (18) for t < V /W and t ≥ V /W , respectively. 4.2.2

Boundedness hypothesis

We assume throughout this subsection that Gi ∧ EK,i holds. We see first that (12) is trivial for YK+ , recalling a given open pair lies in at most k open triples and so YK+ (i) ≤ k < k ·

n1/3 S2 = 2 ε. 7ε n s2 kn

We turn now to YK− . As K is uncovered, each vertex has at most one neighbour in K in G(i), and each partial triple (uv, w) ∈ YK (i) contains a unique open pair, which we take to be uw without loss of generality. We note that the trivial bound |YK− (i)| ≤ |Cei+1 (i)| + 1 does not suffice, as for most steps i, |Cei+1 (i)| = n2/3+o(1) , while the required upper bound S2 /(s22 knε ) = n2/3−7ε+o(1) , so we must be more careful. Suppose ei+1 = xy, and let ∆i = ∆(G(i)). As a triple (uv, w) ∈ YK (i) is counted by YK− (i) if and only if uw ∈ {xy} ∪ Cxy (i), let A = {(uv, w) ∈ YK (i) : uw ∈ Cxy (i)}. We partition A = A1 ∪ A2 and bound each part separately, where A1 = {(uv, w) ∈ A : u ∈ / {x, y}}

and

A2 = {(uv, w) ∈ A : u ∈ {x, y}} .

We first claim |A1 | ≤ 6∆i : at most 4∆i triples (uv, w) ∈ A1 satisfy w ∈ / {x, y}, as then u ∈ (Ni (x) ∪ Ni (y)) ∩ K and w ∈ Ni (x) ∪ Ni (y). Similarly, at most 2∆i such triples satisfy w ∈ {x, y}, as if, say, w = x, then some z ∈ [n] satisfies {yz, zu} ⊆ E(i), and the choice of z ∈ Ni (y) determines a unique u. Next, suppose (uv, w) ∈ A2 and u = x, so there exists a zw ∈ [n] such that {yzw , zw w} ⊆ E(i). Trivially, at most ∆i such triples satisfy v = y, and at most ∆i satisfy v = zw , as then zw is y’s unique neighbour in K. If, instead, v ∈ / {y, zw }, then the sequence (y, zw , w, v) forms a path of length 3 from y into K in G(i), and Theorem 4 yields the number of such triples is at most k · n12ε . Combining these bounds and accounting for the analogous case u = y yields |A2 | ≤ 4∆i + 2kn12ε . As ε < 1/57 and as ∆i = o(kn12ε ) by Theorem 4, (12) follows for YK− as YK− (i) ≤ 1 + |A| ≤ 1 + 10∆i + 2kn12ε ≤ kn1/3−7ε =

15

S2 . 2 s2 knε

4.3

Analytic considerations

Here we verify the remaining inequalities from Part 4. of Lemma 8, recalling that we chose c = 1/4 and C sufficiently large. First, from (1), we have tmax = µ(log n)1/3 = o(nε/2 ) so for large n, we have n3ε < s < tmax s = m < nε/2 s ≤ n2 . For any fixed constant C > 0, j ∈ {1, 2}, and n sufficiently large, as kj ≤ n1/3+ε , 40Cs2j kj nε ≤ 40Cn1/3+8ε < n1/3+9ε < s. As x(t), y(t), e(t), γ(t) are nonnegative, it follows from the definitions that, for j ∈ {1, 2}, inf θj (t) + t≥0

1 γ(t) 1 ej (t)xj (t) γj (t) − ≥ inf + > . t≥0 2 2 2 2 4

Next, we observe from the definitions of x, y, x± , y ± and straightforward differentiation that we can bound, for t ≥ 0, |x0 |, |y 0 |, |x± |, |y ± |, |x00 |, and |y 00 | above by a function of the 3 form H(t) = F (t)e−8t , where R ∞ F is a polynomial of degree 5 with nonnegative coefficients. As clearly supt≥0 H(t) and 0 H(t) dt are finite, any choice of C than both provides R ∞greater 00 0 (t)| dt. (t)|, and |x the required upper bound on supt≥0 |x± (t)|, sup |x t≥0 j j j 0 0 Finally, turning R ∞to the inequalities involving hj = (ej xj + γj ) , it is easy to see that supt≥0 |γ 0 (t)| and 0 |γ 00 (t)| are bounded. Calculations similar to those establishing lower bounds on h1 , h2 easily yield that |(ej xj )0 | and |(ej xj )00 | are bounded above by a function of the form F (t)eP (t) , where F is a polynomial of degree 5 with nonnegative coefficients. As m/s = tmax , eP (t) < (eP (t) )0 , and F (t) and eP (t) are increasing, it follows that Z tmax F (t)eP (t) dt ≤ F (tmax )eP (tmax ) = sup F (t)eP (t) . 0≤t≤tmax

0

Since F (tmax ) = o(nε/2 ) and since eP (t) ≤ nε/2 on [0, tmax ], the bounds sup0≤t≤m/s |hj (t)| < nε R m/s and 0 |h0j (t)| dt < nε follow, and the proof of Lemma 6 is complete.

Acknowledgement The author wishes to thank the anonymous referees for many helpful suggestions which resulted in numerous simplifications and improved the presentation of this paper.

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