Turan numbers of complete 3-uniform Berge-hypergraphs

Abstract

Given a family $\mathcal{F}$ of $r$-graphs, the Tur\'{a}n 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 $r\geq 3$, 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 $e_{ij},$ $1\leq i<j\leq n,$ 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 $n \geq 13$. We also find the extremal hypergraphs avoiding $\mathcal{F}^{(3)}_{n}$.