On regular hypergraphs of high girth

Abstract

We give lower bounds on the maximum possible girth of an $r$-uniform, $d$-regular hypergraph with at most $n$ vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between $3/2+o(1)$ and $2 +o(1)$). We also define a random $r$-uniform `Cayley' hypergraph on $S_n$ which has girth $Ω(n^{1/3})$ with high probability, in contrast to random regular $r$-uniform hypergraphs, which have constant girth with positive probability.

A hole in the proof of Theorem 8 was recently pointed out by Eberhard [10]. In this version, we repair the hole, giving a slightly weaker bound. We use a very similar method to that of Eberhard in [10], where a slightly weaker variant of [13, Theorem 3] is proved; our original proof contained a similar hole to the original proof of [13, Theorem 3]