Turán Numbers for Forests of Paths in Hypergraphs

Abstract

The Turán number of an $r$-uniform hypergraph $H$ is the maximum number of edges in any $r$-graph on $n$ vertices which does not contain $H$ as a subgraph. Let $\mathcal{P}_{\ell}^{(r)}$ denote the family of $r$-uniform loose paths on $\ell$ edges, $\mathcal{F}(k,l)$ denote the family of hypergraphs consisting of $k$ disjoint paths from $\mathcal{P}_{\ell}^{(r)}$, and $L_\ell^{(r)}$ denote an $r$-uniform linear path on $\ell$ edges. We determine precisely ${ex}_r(n;\mathcal{F}(k,l))$ and ${ex}_r(n;k\cdot L_\ell^{(r)})$, as well as the Turán numbers for forests of paths of differing lengths (whether these paths are loose or linear) when $n$ is appropriately large dependent on $k,l,r$ for $r\geq 3$. Our results build on recent results of Füredi, Jiang, and Seiver, who determined the extremal numbers for individual paths, and provide more hypergraphs whose Turán numbers are exactly determined.