  For which graphs $F$ is there a sparse $F$-counting lemma in $C_4$-free
graphs? We are interested in identifying graphs $F$ with the property that,
roughly speaking, if $G$ is an $n$-vertex $C_4$-free graph with on the order of
$n^{3/2}$ edges, then the density of $F$ in $G$, after a suitable
normalization, is approximately at least the density of $F$ in an
$\epsilon$-regular approximation of $G$. In recent work, motivated by
applications in extremal and additive combinatorics, we showed that $C_5$ has
this property. Here we construct a family of graphs with the property.
