Which graphs can be counted in $C_4$-free graphs?

TL;DR

This paper investigates which graphs can be counted in $C_4$-free graphs using a sparse counting lemma, extending previous results by identifying a new family of such graphs.

Contribution

The authors construct a new family of graphs for which a sparse $F$-counting lemma holds in $C_4$-free graphs, broadening the class of graphs with this property.

Findings

Established a family of graphs with the sparse counting property in $C_4$-free graphs

Extended previous work from $C_5$ to a broader class of graphs

Provided insights into graph counting in extremal combinatorics

Abstract

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.