Hereditary quasirandomness without regularity

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.