Linear dependence between hereditary quasirandomness conditions

TL;DR

This paper proves that a hereditary quasirandomness condition involving counts of a fixed graph H in specific vertex subsets implies the graph is quasirandom, with a linear relationship between parameters, extending previous results.

Contribution

It establishes a linear dependence of the quasirandomness parameter on the hereditary condition for any nonempty graph H, generalizing prior work limited to complete graphs.

Findings

Hereditary counts imply quasirandomness with linear parameter dependence.

Extension of results from complete graphs to arbitrary graphs.

Provides a new framework for understanding quasirandomness conditions.

Abstract

Answering a question of Simonovits and S\' os, Conlon, Fox, and Sudakov proved that for any nonempty graph $H$, and any $\varepsilon>0$, there exists $\delta>0$ polynomial in $\varepsilon$, such that if $G$ is an $n$-vertex graph with the property that every $U\subseteq V(G)$ contains $p^{e(H)}|U|^{v(H)}\pm\delta n^{v(H)}$ labeled copies of $H$, then $G$ is $(p,\varepsilon)$-quasirandom in the sense that every subset $U\subseteq G$ contains $\frac{1}{2}p|U|^{2}\pm\varepsilon n^{2}$ edges. They conjectured that $\delta$ may be taken to be linear in $\varepsilon$ and proved this in the case that $H$ is a complete graph. We study a labelled version of this quasirandomness property proposed by Reiher and Schacht. Let $H$ be any nonempty graph on $r$ vertices $v_{1},\ldots,v_{r}$, and $\varepsilon>0$. We show that there exists $\delta=\delta(\varepsilon)>0$ linear in $\varepsilon$, such that if $G$ is an $n$-vertex graph with the property that every sequence of $r$ subsets $U_{1},\ldots,U_{r}\subseteq V(G)$, the number of copies of $H$ with each $v_{i}$ in $U_{i}$ is $p^{e(H)}\prod|U_{i}|\pm\delta n^{v(H)}$, then $G$ is $(p,\varepsilon)$-quasirandom.