Hypergraph extensions of the Erdős-Gallai Theorem

Abstract

We extend the Erdős-Gallai Theorem for Berge paths in r-uniform hypergraphs. We also find the extremal hypergraphs avoiding t-tight paths of a given length and consider this extremal problem for other definitions of paths in hypergraphs.

Linear Turán numbers of acyclic triple systems

“…In the case of the path, the extension of the Turán problem to hypergraphs was studied for several different notions of paths. For loose paths Mubayi and Verstraëte [17], 2007, for linear paths Füredi, Jiang and Seiver [10], 2014, and for Berge paths Győri, Katona and Lemons [15], 2016, determined exactly (at least for large n and fixed k) the Turán number. It does not seem easy to find the asymptotic of ex L (n, P k ).…”

Tight Cycles in Hypergraphs

“…An approximate version of the analogue of Dirac's theorem was proved recently by Rödl, Ruciński and Szemerédi [10] (and the same authors proved an exact bound for 3-uniform hypergraphs in [11]). Towards a hypergraph analogue of the Erdős-Gallai Theorem, Győri, Katona and Lemons [6] proved that a k-uniform hypergraph on n vertices with more than (αn − k) n k−1 edges contains a tight path on αn vertices. Our first main result improves upon this by approximately a factor of k. Theorem 1.…”

Induced Turán problems and traces of hypergraphs

“…r is the family of r-element vertex sets that span a K r in H. In particular Ever since Győri, G. Y. Katona, and Lemons [8] investigated hypergraphs without long Berge paths there is a renewed interest concerning extremal Berge type problems. Here we define a related function, the induced Berge Turán number of F .…”