On hypergraphs of girth five (Q1408529)

Summary: We study \(r\)-uniform hypergraphs \(\mathcal H\) 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 \(\mathcal H\) has \(n\) vertices and a maximum number of edges, then \[ |\mathcal H |=\frac 16n^{3/2}+o(n^{3/2}). \] This also asymptotically determines the generalized Turán number \(T_3(n,8,4)\). Some results are based on our bounds for the maximum size of Sidon-type sets in \(\mathbb{Z}_n\).