Turán Numbers of Complete 3-Uniform Berge-Hypergraphs

Abstract

Given a family ${\mathcal {F}}$ of r-graphs, the Turán number of ${\mathcal {F}}$ for a given positive integer N, denoted by $ex(N,{\mathcal {F}})$, is the maximum number of edges of an r-graph on N vertices that does not contain any member of ${\mathcal {F}}$ as a subgraph. For given $r\ge 3$, a complete r-uniform Berge-hypergraph, denoted by ${K}_n^{(r)}$, is an r-uniform hypergraph of order n with the core sequence $v_{1}, v_{2}, \ldots ,v_{n}$ as the vertices and distinct edges $e_{ij},$ $1\le i<j\le n,$ where every $e_{ij}$ contains both $v_{i}$ and $v_{j}$. Let ${\mathcal {F}}^{(r)}_n$ be the family of complete r-uniform Berge-hypergraphs of order n. We determine precisely $ex(N,{\mathcal {F}}^{(3)}_{n})$ for $N \ge n \ge 13$. We also find the extremal hypergraphs avoiding ${\mathcal {F}}^{(3)}_{n}$.