Turán Numbers for Forests of Paths in Hypergraphs

Neal Bushaw, Nathan Kettle

Published 2013-03-20, updated 2014-02-24Version 4

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.