The typical structure of graphs with no large cliques

Abstract

In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed r ∈ N, almost every n-vertex K r+1 -free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r (log n) 1/4 . The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobás and Simonovits.

On the Number of Graphs Without Large Cliques

“…Note also that, similarly to the result of Erdős, Kleitman and Rothschild [8], our theorem just provides the asymptotics of the logarithm of f n (K ℓn ). However, our paper has already stimulated further research, and very recently a structural result in the spirit of Kolaitis, Prömel and Rothschild has been established by Balogh et al [3].…”

The Method of Hypergraph Containers

“…Proving such a result is less straightforward than Lemma 2.3; for example, one natural proof combines the triangle removal lemma of Ruzsa and Szemerédi [75] with the classical stability theorem of Erdős and Simonovits [33,86]. However, an extremely simple, beautiful, and elementary proof was given recently by Füredi [45] (see also [8]). Lemma 2.5 (Robust stability for triangles).…”

The Typical Structure of Gallai Colorings and Their Extremal Graphs

“…Assume that P is a Gallai 3-template of G with |Ga(P, G)| > 2 (1−δ)( n 2 ) . Then there exist two colors i, j ∈ [3] such that the number of edges of K n with palette {i, j} is at least (1−37·40δ) n 2 .…”

“…We say that a graph G is t-far from being k-partite if χ(G ′ ) > k for every subgraph G ′ ⊂ G with e(G ′ ) > e(G) − t. We will use the following theorem of Balogh, Bushaw, Collares, Liu, Morris, and Sharifzadeh [3]. Theorem 6.13.…”

“…Theorem 6.13. [3] For every n, k, t ∈ N, the following holds. Every graph G of order n which is t-far from being k-partite contains at least n k−1 e 2k · k!…”