A new proof of the K\L R conjecture

TL;DR

This paper provides a new direct proof of the K extbar L R conjecture, a fundamental result in random graph theory related to the probability of Erd extbar os-Rényi graphs being H-free, using induction instead of hypergraph containers.

Contribution

It introduces a novel inductive proof approach for the K extbar L R conjecture, simplifying the understanding of the probability bounds in random graphs.

Findings

Established a new proof technique for the K extbar L R conjecture

Demonstrated the superexponential smallness of H-free probability under uniform edge distribution

Connected the proof to implications in the sparse regularity lemma

Abstract

Estimating the probability that the Erd\H{o}s-R\'enyi random graph $G(n,m)$ is $H$-free, for a fixed graph $H$, is one of the fundamental problems in random graph theory. If $m$ is such that each edge of $G(n,m)$ belongs to a copy of $H'$ for every $H' \subseteq H$, in expectation, then it is known that $G(n,m)$ is $H$-free with probability $\exp(- \Theta(m))$. The KLR 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.