Turán Numbers of Complete 3-Uniform Berge-Hypergraphs

Abstract

Given a family $${\mathcal {F}}$$ of r-graphs, the Turan 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 $${\mathcal {F}}^{(3)}_{n}$$ .