Hereditary quasirandomness without regularity

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

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