Hypergraph Turán Numbers of Vertex Disjoint Cycles

Abstract

The Turán number of a k-uniform hypergraph H, denoted by ex_k(n;H), is the maximum number of edges in any k-uniform hypergraph F on n vertices which does not contain H as a subgraph. Let ${\cal C}_\ell ^{(k)}$ denote the family of all k-uniform minimal cycles of length ℓ, ${\cal S}({\ell _1}, \cdots ,{\ell _r})$ denote the family of hypergraphs consisting of unions of r vertex disjoint minimal cycles of length ℓ₁,…,ℓ_r, respectively, and $\mathbb{C}_\ell ^{(k)}$ denote a k-uniform linear cycle of length ℓ. We determine precisely $e{x_k}\left( {n;{\cal S}({\ell _1}, \cdots ,{\ell _r})} \right)$ and $e{x_k}\left( {n;\mathbb{C}_{{\ell _1}}^{(k)}, \ldots ,\mathbb{C}_{{\ell _r}}^{(k)}} \right)$ for sufficiently large n. Our results extend recent results of Füredi and Jiang who determined the Turán numbers for single k-uniform minimal cycles and linear cycles.