The largest minimum codegree of a 3-graph without a generalized 4-cycle

摘要

A 3-graph is said to contain a generalized 4-cycle if it contains 4 edges A,B,C,D such that A ∩ B = C ∩ D = ? and A ∪ B = C ∪ D. We show that a 3-graph in which every pair of vertices is contained in at least 4 edges must contain a generalized 4-cycle. When the number of vertices, n, is equivalent to 1 or 5 modulo 20, this result is optimum, in the sense that for such n there are 3-graphs where every pair of vertices is contained in 3 edges but which do not contain a generalized 4-cycle.