Hypergraphs with No Cycle of a Given Length

Abstract

Recently, the authors gave upper bounds for the size of 3-uniform hypergraphs avoiding a given odd cycle using the definition of a cycle due to Berge. In the present paper we extend this bound to m -uniform hypergraphs (for all m ≥ 3), as well as m -uniform hypergraphs avoiding a cycle of length 2 k . Finally we consider non-uniform hypergraphs avoiding cycles of length 2 k or 2 k + 1. In both cases we can bound | h | by O ( n 1+1/ k ) under the assumption that all h ∈ ε( ) satisfy | h | ≥ 4 k 2 .