  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.
