Avoiding long Berge cycles: the missing cases k = r + 1 and k = r + 2

Abstract

The maximum size of an r-uniform hypergraph without a Berge cycle of length at least k has been determined for all k ≥ r + 3 by Füredi, Kostochka and Luo and for k < r (and k = r, asymptotically) by Kostochka and Luo. In this paper, we settle the remaining cases: k = r + 1 and k = r + 2, proving a conjecture of Füredi, Kostochka and Luo.Recently, Füredi, Kostochka and Luo [3] proved exact bounds similar to Theorem 1 for hypergraphs avoiding long Berge cycles.

Berge Cycles in Non-Uniform Hypergraphs

“…There are many exact results concerning the maximum size of uniform hypergraphs avoiding Berge paths and cycles, see the recent results of Ergemlidze et al [7] or one by the present authors [8].…”

General lemmas for Berge–Turán hypergraph problems

“…When k ≤ r we give the following result without assuming that the Erdős-Sós conjecture is true. Our proof of the theorem below is inspired by some ideas in [8].…”

“…They also prove the unexpected result that the maximum number of hyperedges in an n-vertex r-uniform Berge-C 2k+1 -free hypergraph (for r ≥ 3) is also O(n 1+1/k ) which is significantly different from the graph case. Very recently, the problem of avoiding all Berge cycles of length at least k has been investigated in a series of papers [12,8,23]. For general results on the maximum size of a Berge-F -free hypergraph for an arbitrary graph F see Gerbner and Palmer [15] and Grósz, Methuku and Tompkins [17].…”

The structure of hypergraphs without long Berge cycles

“…Later they [6] also determined exact bounds and extremal constructions for all n, for the case k ≥ r+4. Kostochka and Luo [9] determine a bound for k ≤ r − 1 which is sharp for infinitely many n. Ergemlidze, Győry, Metukhu, Salia, Tompikns and Zamora [4] determine a bound in the cases where k ∈ {r + 1, r + 2}. The case when k = r remained open.…”