Turán Numbers of Complete 3-Uniform Berge-Hypergraphs

Abstract

Given a family F of r-graphs, the Turán number of F for a given positive integer N, denoted by ex(N, F) , is the maximum number of edges of an r-graph on N vertices that does not contain any member of F as a subgraph. For given r≥ 3 , a complete r-uniform Berge-hypergraph, denoted by Kn(r), is an r-uniform hypergraph of order n with the core sequence v1, v2, … , vn as the vertices and distinct edges eij, 1 ≤ i< j≤ n, where every eij contains both vi and vj. Let Fn(r) be the family of complete r-uniform Berge-hypergraphs of order n. We determine precisely ex(N,Fn(3)) for N≥ n≥ 13. We also find the extremal hypergraphs avoiding Fn(3). © 2018, Springer Japan KK, part of Springer Nature.