Avoiding long Berge cycles

Abstract

Let $n\ge k\ge r+3$ and $\mathcal{H}$ be an $n$-vertex $r$-uniform hypergraph. We show that if $|\mathcal{H}|>\frac{n-1}{k-2}\left(\genfrac{}{}{0ex}{}{k-1}{r}\right)$ then $\mathcal{H}$ contains a Berge cycle of length at least $k$. This bound is tight when $k-2$ divides $n-1$. We also show that the bound is attained only for connected $r$-uniform hypergraphs in which every block is the complete hypergraph $K_{k-1}^{(r)}$. We conjecture that our bound also holds in the case $k=r+2$, but the case of short cycles, $k\le r+1$, is different.