On regular hypergraphs of high girth

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 the symmetric group S_n which has girth Ω(√ log |S_n|) with high probability, in contrast to random regular r-uniform hypergraphs, which have constant girth with positive probability.