Turan 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 $v_1,v_2,\dots ,v_n$ 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 ${\mathcal{F}}_n^{(r)}$ be the family of complete $r$-uniform Berge-hypergraphs of order $n.$ We determine precisely $ex(N,{\mathcal{F}}_n^{(3)})$ for $n\ge 13$. We also find the extremal hypergraphs avoiding ${\mathcal{F}}_n^{(3)}$.