Turan Numbers for Forests of Paths in Hypergraphs

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 P[(r) over l] denote the family of r-uniform loose paths on l edges, F(k,l) denote the family of hypergraphs consisting of k disjoint paths from P[(r) over l], and L[(r) over l] denote an r-uniform linear path on l edges. We determine precisely ex_r(n; F(k,l)) and ex_r(n; . L[(r) over l]), 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 ≥ 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.