Combinatorial theorems in sparse random sets

TL;DR

This paper introduces a unified technique to prove that classical combinatorial theorems like Turán's, Szemerédi's, and Ramsey's hold almost surely in sparse random sets, extending known results to the random setting.

Contribution

It develops a new method that unifies the proof of multiple combinatorial theorems in sparse random environments, including sharp thresholds and structural results.

Findings

Extended Turán's theorem to sparse random graphs with sharp thresholds

Proved sparse analogues of structural results like stability and hypergraph removal lemmas

Established that many classical combinatorial theorems hold almost surely in sparse random sets

Abstract

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\'an's theorem, Szemer\'edi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For instance, we extend Tur\'an's theorem to the random setting by showing that for every $\epsilon > 0$ and every positive integer $t \geq 3$ there exists a constant $C$ such that, if $G$ is a random graph on $n$ vertices where each edge is chosen independently with probability at least $C n^{-2/(t+1)}$, then, with probability tending to $1$ as $n$ tends to infinity, every subgraph of $G$ with at least $(1 - \frac{1}{t-1} + \epsilon) e(G)$ edges contains a copy of $K_t$. This is sharp up to the constant $C$. We also show how to prove sparse analogues of structural results, giving two main applications, a stability version of the random Tur\'an theorem stated above and a sparse hypergraph removal lemma. Many similar results have recently been obtained independently in a different way by Schacht and by Friedgut, R\"odl and Schacht.