Turan Numbers For Forests Of Paths In Hypergraphs

The Turan 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-l((r)) 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-l((r)), and L-l((r)) denote an r-uniform linear path on l edges. We determine precisely ex(r)(n; F(k, l)) and ex(r)(n; k L-l((r))), as well as the Turan 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 Furedi, Jiang, and Seiver, who determined the extremal numbers for individual paths, and provide more hypergraphs whose Turan numbers are exactly determined.