Pancyclicity in hypergraphs with large uniformity

Abstract

A Berge cycle of length $\ell$ in a hypergraph $\mathcal{H}$ is a sequence of alternating vertices and edges $v_0{e}_0{v}_1{e}_1...v_{\ell }{e}_{\ell }{v}_0$ such that $\{v_i,v_{i+1}\}\subseteq e_i$ for all $i$, with indices taken modulo $\ell$. For $n$ sufficiently large and $r\ge \lfloor \frac{n-1}{2}\rfloor -1$ we prove exact minimum degree conditions for an $n$-vertex, $r$-uniform hypergraph to contain Berge cycles of every length between $2$ and $n$. In conjunction with previous work, this provides sharp Dirac-type conditions for pancyclicity in $r$-uniform hypergraphs for all $3\le r\le n$ when $n$ is sufficiently large.