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arXiv · 2106.03261

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

David Conlon·Jacob Fox·Benny Sudakov·Yufei Zhao
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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.

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David Conlon, Jacob Fox, Benny Sudakov, Yufei Zhao. 2021-06-06. Which graphs can be counted in $C_4$-free graphs?. https://arxiv.org/abs/2106.03261

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