A new proof of the KŁR conjecture

Abstract

Estimating the probability that the Erdős-Rényi random graph Gn,m is H-free, for a fixed graph H, is one of the fundamental problems in random graph theory. If H is non-bipartite and m is such that each edge of Gn,m belongs to a copy of H′ for every H′⊆H, in expectation, then it is known that Gn,m is H-free with probability exp⁡(−Θ(m)). The KŁR conjecture, slightly rephrased, states that if we further condition on uniform edge distribution, the archetypal property of random graphs, the probability of being H-free becomes superexponentially small in the number of edges. While being interesting on its own, the conjecture has received significant attention due to its connection with the sparse regularity lemma, and the many results in random graphs that follow. It was proven by Balogh, Morris, and Samotij and, independently, by Saxton and Thomason, as one of the first applications of the hypergraph containers method. We give a new direct proof using induction.