The number of B3-sets of a given cardinality

Dellamonica, Domingos; Kohayakawa, Yoshiharu; Lee, Sang June; Rödl, Vojtěch; Samotij, Wojciech

doi:10.1016/j.jcta.2016.03.004

Cited by 9 publications

(13 citation statements)

References 15 publications

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“…In Section 7 we shall prove a similar (and also probably close to best possible) theorem for K s,t -free graphs. We remark that similar results in the closely related setting of B hsets (though using somewhat different techniques) were also obtained recently in a series of papers by Dellamonica, Kohayakawa, Lee, Rödl and Samotij [18,19,20,38].…”

Section: Introductionsupporting

confidence: 78%

The number of C2-free graphs

Morris

Saxton

2016

Advances in Mathematics

124

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One of the most basic questions one can ask about a graph H is: how many H-free graphs on n vertices are there? For non-bipartite H, the answer to this question has been well-understood since 1986, when Erdős, Frankl and Rödl proved that there are 2 (1+o(1))ex(n,H) such graphs. For bipartite graphs, however, much less is known: even the weaker bound 2 O(ex(n,H)) has been proven in only a few special cases: for cycles of length four and six, and for some complete bipartite graphs.For even cycles, Bondy and Simonovits proved in the 1970s that ex(n, C 2ℓ ) = O n 1+1/ℓ , and this bound is conjectured to be sharp up to the implicit constant. In this paper we prove that the number of C 2ℓ -free graphs on n vertices is at most 2 O(n 1+1/ℓ ) , confirming a conjecture of Erdős. Our proof uses the hypergraph container method, which was developed recently (and independently) by Balogh, Morris and Samotij, and by Saxton and Thomason, together with a new 'balanced supersaturation theorem' for even cycles. We moreover show that there are at least 2 (1+c)ex(n,C6) C 6 -free graphs with n vertices for some c > 0 and infinitely many values of n ∈ N, disproving a well-known and natural conjecture. As a further application of our method, we essentially resolve the so-called Turán problem on the Erdős-Rényi random graph G(n, p) for both even cycles and complete bipartite graphs.Research supported in part by a CNPq bolsa PDJ (DS) and by CNPq Proc. 479032/2012-2 and Proc. 303275/2013-8 (RM). 1 1.1. History and background. The study of extremal graph theory was initiated roughly 70 years ago by Turán [52], who determined exactly the extremal number of the complete graph, by Erdős and Stone [29], who determined asymptotically (for all r 3) the extremal number of a complete r-partite graph 1 , and by Kővári, Sós and Turán [41] who showed that ex(n, K s,t ) = O(n 2−1/s ), where K s,t denotes the complete bipartite graph with part sizes s and t. (The case K 2,2 = C 4 was solved some years earlier by Erdős [21] during his study of multiplicative Sidon sets.) Over the following decades, a huge amount of effort was put into determining more precise bounds for specific families of graphs (see, e.g., [11,31]), and a great deal of progress has been made. Nevertheless, the order of magnitude of ex(n, H) for most bipartite graphs, including simple examples such as K 4,4 and C 8 , remains unknown.In the 1970s, the problem of determining the number of H-free graphs on n vertices was introduced by Erdős, Kleitman and Rothschild [23], who proved that there are 2 (1+o(1))ex(n,Kr) K r -free graphs, and moreover that almost all triangle-free graphs are bipartite. This latter result was extended to all cliques by Kolaitis, Prömel and Rothschild [39] and to more general graphs by Prömel and Steger [46], and the former to all non-bipartite graphs by Erdős, Frankl and Rödl [24], using Szemerédi's regularity lemma. The corresponding result for k-uniform hypergraphs was proved by Nagle, Rödl and Schacht [44] using hypergraph regularity, and reproved by Balogh,...

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Section: Introductionsupporting

confidence: 78%

The number of C2-free graphs

Morris

Saxton

2016

Advances in Mathematics

124

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“…We postpone the (fairly straightforward) derivation of Theorem to § 5.2 and focus on Theorem instead. First, we need several additional definitions which generalize the concepts already introduced in .…”

Section: Proof Outlinementioning

confidence: 99%

“…Since there seems to be no easy way of controlling

false∥ R_{scriptG} false∥

‘directly’, similarly as in , we shall instead maintain an upper bound on the moment generating function of

R_{G}

, defined as follows. Definition Given a

k

‐graph

scriptG

and an

ℓ

‐graph

scriptH

with

V (G), V (H) \subset [n]

and a positive real

λ

, we let

0true Q_{scriptG, scriptH} (λ) = \sum_{z = - ℓ n}^{k n} prefixexp (() λ \cdot R_{G, H} false(z false)) .

Note that the range of the above sum includes all

z

for which

R_{G, H} false(z false) \neq 0

.…”

Section: Proof Outlinementioning

confidence: 99%

The number of Bh‐sets of a given cardinality

Dellamonica

Kohayakawa

Lee

et al. 2017

Proc. London Math. Soc.

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For any integer h⩾2, a set A of integers is called a Bh‐set if all sums a1+⋯+ah, with a1,…,ah∈A and a1⩽⋯⩽ah, are distinct. We obtain essentially sharp asymptotic bounds for the number of Bh‐sets of a given cardinality that are contained in the interval {1,⋯,n}. As a consequence of these bounds, we determine, for any integer m⩽n, the cardinality of the largest Bh‐set contained in a typical m‐element subset of {1,…,n}.

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“…Concerning the question of the maximum size of Sidon or B h sets in infinite random sets of integers, Theorem 1.4 currently gives the best lower bound when h = 2 and 5/6 < δ < 1. In the case of h > 2 no other bounds are known, though we believe that using results of Dellamonica et al [4,5] for the case of finite random sets, one can establish exact exponents whenever 0 < δ < h/(2h − 1). Note that Kohayakawa et al [9] also made use of the case of finite random sets established in [11].…”

Section: Remarks and Open Questionsmentioning

confidence: 93%

On strong infinite Sidon and $B_h$ sets and random sets of integers

Fabian¹,

Rué²,

Spiegel³

2019

Preprint

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A set of integers S ⊂ N is an α-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on α, more specifically if (x + w) − (y + z) ≥ max{x α , y α , z α , w α } for every x, y, z, w ∈ S satisfying max{x, w} = max{y, z}. We obtain a new lower bound for the growth of α-strong infinite Sidon sets when 0 ≤ α < 1. We also further extend that notion in a natural way by obtaining the first non-trivial bound for α-strong infinite B h sets.In both cases, we study the implications of these bounds for the density of, respectively, the largest Sidon or B h set contained in a random infinite subset of N. Our theorems improve on previous results by Kohayakawa, Lee, Moreira and Rödl.

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The number of B3-sets of a given cardinality

Cited by 9 publications

References 15 publications

The number of C2-free graphs

The number of C2-free graphs

The number of Bh‐sets of a given cardinality

On strong infinite Sidon and $B_h$ sets and random sets of integers

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