On the K{\L}R conjecture in random graphs

TL;DR

This paper proves a variant of the K{ extL}R conjecture in random graphs, enabling the derivation of probabilistic versions of classical extremal combinatorial theorems and broadening applications in random graph theory.

Contribution

The authors establish a variant of the K{ extL}R conjecture applicable to most known applications, facilitating probabilistic versions of extremal combinatorial results.

Findings

Proved a variant of the K{ extL}R conjecture for random graphs.

Derived probabilistic versions of classical extremal theorems.

Discussed further applications in random graph theory.

Abstract

The K{\L}R conjecture of Kohayakawa, {\L}uczak, and R\"odl 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 R\"odl. 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.