Supersaturated Sparse Graphs and Hypergraphs

Abstract

A central problem in extremal graph theory is to estimate, for a given graph H, the number of H-free graphs on a given set of n vertices. In the case when H is not bipartite, fairly precise estimates on this number are known. In particular, thirty years ago, Erdős, Frankl, and Rödl proved that there are 2 (1+o(1))ex(n,H) such graphs. In the bipartite case, however, nontrivial bounds have been proven only for relatively few special graphs H.We make a first attempt at addressing this enumeration problem for a general bipartite graph H. We show that an upper bound of 2 O(ex(n,H)) on the number of H-free graphs with n vertices follows merely from a rather natural assumption on the growth rate of n → ex(n, H); an analogous statement remains true when H is a uniform hypergraph. Subsequently, we derive several new results, along with most previously known estimates, as simple corollaries of our theorem. At the heart of our proof lies a general supersaturation statement that extends the seminal work of Erdős and Simonovits. The bounds on the number of H-free hypergraphs are derived from it using the method of hypergraph containers.

The Method of Hypergraph Containers

“…and k ∈ {2, 3, 5}, since in these cases it is known that ex(n, K s,t ) = Θ(n 2−1/s ) and ex(n, C 2k ) = Θ(n 1+1/k ). Very recently, Ferber, McKinley, and Samotij [38], inspired by a similar result of Balogh, Liu, and Sharifzadeh [10] on sets of integers with no k-term arithmetic progression, found a very simple proof of the following much more general theorem. Note that Theorem 4.2 resolves Conjecture 4.1 for every H such that ex(n, H) = Θ(n α ) for some constant α.…”

“…Note that Theorem 4.2 resolves Conjecture 4.1 for every H such that ex(n, H) = Θ(n α ) for some constant α. Moreover, it was shown in [38] that the weaker assumption that ex(n, H) ≫ n 2−1/m 2 (H)+ε for some ε > 0 already implies that the assertion of Conjecture 4.1 holds for infinitely many n; we refer the interested reader to [38] for details. Let us also note here that, while it is natural to suspect that in fact the stronger bound (8) holds for all graphs H that contain a cycle, this is false for H = C 6 , as was shown by Morris and Saxton [66].…”

“…One easy way to do this is simply to remove one edge from each copy of C4. A more efficient method, used by Kohayakawa, Kreuter, and Steger[58] to improve the lower bound by a polylogarithmic factor, utilizes a version of the general result of[1] on independent sets in hypergraphs obtained in[32]; see also[38].…”

Balanced supersaturation for some degenerate hypergraphs

“…Theorems 1.3 and 1.4 follow immediately from Proposition 1.6 combined with Theorems 1.7 and 1.8. The subsequent work of Ferber et al [17] establishes a weaker, but significantly easier to prove, supersaturation result that is still sufficiently strong to derive F n H ( , ) = 2 r n H ex ( , ) r for a much larger class of r-graphs H . However, the result of [17] is not strong enough to imply anything nontrivial for the Turán problem in random hypergraphs.…”

Independent Sets in Hypergraphs and Ramsey Properties of Graphs and the Integers

“…in the case when H is bipartite; see e.g. [22,48]. Given any k, r, n ∈ N with k ≥ 2 and kuniform hypergraphs H 1 , .…”