Hereditary quasirandomness without regularity

TL;DR

This paper provides a new proof for a quasirandomness result in graphs that avoids the regularity lemma, showing improved bounds on parameters for clique and general graphs.

Contribution

It offers an alternative proof to a known theorem, removing the regularity lemma and establishing linear and polynomial bounds for specific graph classes.

Findings

Avoids the regularity lemma in the proof

Establishes linear bounds for cliques

Provides polynomial bounds for general graphs

Abstract

A result of Simonovits and S\'os states that for any fixed graph $H$ and any $\epsilon > 0$ there exists $\delta > 0$ such that if $G$ is an $n$-vertex graph with the property that every $S \subseteq V(G)$ contains $p^{e(H)} |S|^{v(H)} \pm \delta n^{v(H)}$ labeled copies of $H$, then $G$ is quasirandom in the sense that every $S \subseteq V(G)$ contains $\frac{1}{2} p |S|^2 \pm \epsilon n^2$ edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on $\delta^{-1}$ which is a tower of twos of height polynomial in $\epsilon^{-1}$. We give an alternative proof of this theorem which avoids the regularity lemma and shows that $\delta$ may be taken to be linear in $\epsilon$ when $H$ is a clique and polynomial in $\epsilon$ for general $H$. This answers a problem raised by Simonovits and S\'os.