Uniformity thresholds for the asymptotic size of extremal Berge-$F$-free hypergraphs

TL;DR

This paper investigates the thresholds of hypergraph uniformity for avoiding Berge-$F$ structures, establishing bounds and exact values for these thresholds, including improvements over previous results and connections to chromatic number.

Contribution

It provides new bounds and exact thresholds for the uniformity of hypergraphs avoiding Berge-$F$, including the case of linear hypergraphs and specific graphs like triangles.

Findings

For non-bipartite $F$, $ex_r(n,F)= ext{Omega}(n^2)$ for small $r$, but $o(n^2)$ for large $r$.

The uniformity threshold for triangles is exactly 5, improving previous bounds.

The linear hypergraph threshold equals the chromatic number of $F$.

Abstract

Let $F = (U,E)$ be a graph and $\mathcal{H} = (V,\mathcal{E})$ be a hypergraph. We say that $\mathcal{H}$ contains a Berge-$F$ if there exist injections $\psi:U\to V$ and $\varphi:E\to \mathcal{E}$ such that for every $e=\{u,v\}\in E$, $\{\psi(u),\psi(v)\}\subset\varphi(e)$. Let $ex_r(n,F)$ denote the maximum number of hyperedges in an $r$-uniform hypergraph on $n$ vertices which does not contain a Berge-$F$. For small enough $r$ and non-bipartite $F$, $ex_r(n,F)=\Omega(n^2)$; we show that for sufficiently large $r$, $ex_r(n,F)=o(n^2)$. Let $thres(F) = \min\{r_0 :ex_r(n,F) = o(n^2) \text{ for all } r \ge r_0 \}$. We show lower and upper bounds for $thres(F)$, the uniformity threshold of $F$. In particular, we obtain that $thres(\triangle) = 5$, improving a result of Gy\H{o}ri. We also study the analogous problem for linear hypergraphs. Let $ex^L_r(n,F)$ denote the maximum number of hyperedges in an $r$-uniform linear hypergraph on $n$ vertices which does not contain a Berge-$F$, and let the linear unformity threshold $thres^L(F) = \min\{r_0 :ex^L_r(n,F) = o(n^2) \text{ for all } r \ge r_0 \}$. We show that $thres^L(F)$ is equal to the chromatic number of $F$.