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arXiv · 2606.07126

A note on the extremal number of Berge-$C_4$

Nika Salia·Casey Tompkins
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Abstract

We improve the known upper bound for the extremal number of Berge-$C_4$-free $3$-uniform hypergraphs. More precisely, we prove that every $n$-vertex $3$-uniform hypergraph with no Berge cycle of length four has at most \[ \frac{n^{3/2}}{2+\sqrt2}+O(n) \] hyperedges. This improves the previous best-known leading constant $1/\sqrt{10}$ to $1/(2+\sqrt2)$.

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Nika Salia, Casey Tompkins. 2026-06-05. A note on the extremal number of Berge-$C_4$. https://arxiv.org/abs/2606.07126

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