The structure of hypergraphs without long Berge cycles

Abstract

We study the structure of r-uniform hypergraphs containing no Berge cycles of length at least k for k ≤ r, and determine that such hypergraphs have some special substructure. In particular we determine the extremal number of such hypergraphs, giving an affirmative answer to the conjectured value when k = r and giving a a simple solution to a recent result of Kostochka-Luo when k < r.

Pósa-Type Results for Berge Hypergraphs

“…where Long Berge cycles are well-studied for hypergraphs. Turán-type questions for uniform hypergraphs without long Berge cycles are settled in [8,9,14,23]. Bermond, Germa, Heydemann, and Sotteau [2] found a Dirac-type condition forcing long Berge cycles for uniform hypergraphs.…”

Stability of extremal connected hypergraphs avoiding Berge-paths

“…Later Davoodi, Győri, Methuku and Tompkins [2] settled the missing case r = k + 1. For results on the maximum number of hyperedges in r-uniform hypergraphs not containing Berge-cycles longer than k see [5,10] and the references therein.…”

3-Uniform Hypergraphs without a Cycle of Length Five

“…These bounds were improved by Füredi andÖzkahya [9], Jiang and Ma [19], Gerbner, Methuku and Vizer [11]. Recently Füredi, Kostochka and Luo [7] started the study of the maximum size of an n-vertex r-uniform hypergraph without any Berge cycle of length at least k. This study has been continued in [8,18,20,4].…”