On regular hypergraphs of high girth

TL;DR

This paper establishes lower bounds on the girth of regular hypergraphs and introduces a random hypergraph model with high girth, advancing understanding of hypergraph cycle properties.

Contribution

It provides new lower bounds on hypergraph girth and proposes a random hypergraph construction with high girth, contrasting with typical models.

Findings

Lower bounds differ from trivial upper bounds by a constant factor

A new random hypergraph model achieves girth of order n^{1/3} with high probability

Random regular hypergraphs have constant girth with positive probability

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 $\Omega (n^{1/3})$ with high probability, in contrast to random regular $r$-uniform hypergraphs, which have constant girth with positive probability.