On the KŁR conjecture in random graphs

Abstract

The K LR conjecture of Kohayakawa, Luczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G n,p , for sufficiently large p := p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.Often the strength of the regularity lemma lies in the fact that it may be combined with a counting or embedding lemma that tells us approximately how many copies of a particular subgraph

Extremal Results in Random Graphs

“…In particular, we obtain an appropriate generalization of the embedding lemma from Fact 3.2, for subgraphs of G(n, p) (see Theorem 3.8). This result was also shown by Conlon, Gowers, Samotij, and Schacht [23] directly (without proving Conjecture 3.6). If…”

“…Theorem 3.4 can be proved like the original regularity lemma with fairly straightforward adjustments. To prove a corresponding form of Fact 3.2 turns out to be a challenging problem, which was resolved only recently in [8,23,101]. In particular, in the work of Balogh, Morris, and Samotij [8] and of Saxton and Thomason [101], a conjecture of Kohayakawa, Luczak, and Rödl [65] was addressed.…”

“…This conjecture implies a version of the embedding lemma of Fact 3.2 appropriate for applications of Theorem 3.4. In [23] only such a version was derived (see Theorem 3.8 below). For the formulation of the conjecture from [65], we require some more notation.…”

Sharp thresholds for Ramsey properties of strictly balanced nearly bipartite graphs

“…B,n we start with the family B,n and remove embeddings that do not satisfy property ( 2), embeddings that do not satisfy property ( 4) and embeddings that will later lead to problems for ( 3). After that we choose at random 2 αn 2 embeddings which will induce property ( 3) and show that after deleting the embeddings that intersect in more than one vertex we keep C αn 2 of them with C > 1.…”

“…we select with repetition εn 2 times an element of 3 B,n , where we assume for simplicity that εn 2 is an integer. For every selection S we define a family of embeddings S ⊆ 3…”