Turán Numbers for Forests of Paths in Hypergraphs (1303.5022v4)

Abstract: The Tur\'an 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 P'_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*P'_l^r), as well as the Tur\'an 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 F\"uredi, Jiang, and Seiver who determined the extremal numbers for individual paths, and provide more hypergraphs whose Turan numbers are exactly determined.