On hypergraphs of girth five

Abstract

In this paper, we study r-uniform hypergraphs H without cycles of length less than five, employing the definition of a hypergraph cycle due to Berge. In particular, for r = 3, we show that if H has n vertices and a maximum number of edges, then |H| = 1/6n3/2 + o(n3/2). This also asymptotically determines the generalized Turán number T3(n, 8, 4). Some results are based on our bounds for the maximum size of Sidon-type sets in ℤn.