Tur\'an Numbers for Forests of Paths in Hypergraphs

TL;DR

This paper precisely determines the maximum number of edges in large hypergraphs that avoid containing certain forests of paths, extending previous results to more complex hypergraph structures.

Contribution

It provides exact Turán numbers for forests of paths in r-uniform hypergraphs, including loose and linear paths, for large n, advancing extremal hypergraph theory.

Findings

Exact Turán numbers for forests of paths in hypergraphs

Extension of previous results to more complex path structures

Determination of extremal numbers for large n in various path configurations

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.