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

Ergemlidze, Beka; Győri, Ervin; Methuku, Abhishek; Salia, Nika; Tompkins, Casey; Zamora, Oscar

doi:10.48550/arxiv.1808.07687

Cited by 9 publications

(12 citation statements)

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“…The remaining case of t = k + 1 was settled by Davoodi, Győri, Methuku and Tompkins [5]. For long cycles, Füredi, Kostochka and Luo [7] showed that for k ≥ 3 and t [6]. The case when t = k is recently settled by Győri et al [13].…”

Section: Introductionmentioning

confidence: 98%

On the cover Ramsey number of Berge hypergraphs

Lü¹,

Wang²

2019

Preprint

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For a fixed set of positive integers R, we say H is an R-uniform hypergraph, or Rgraph, if the cardinality of each edge belongs to R. An R-graph H is covering if every vertex pair of H is contained in some hyperedge. For a graph G = (V, E), a hypergraph H is called a Berge-G, denoted by BG, if there exists an injection f : E(G) → E(H) such that for every e ∈ E(G), e ⊆ f (e). In this note, we define a new type of Ramsey number, namely the cover Ramsey number, denoted as RR (BG 1 , BG 2 ), as the smallest integer n 0 such that for every covering R-uniform hypergraph H on n ≥ n 0 vertices and every 2-edge-coloring (blue and red) of H , there is either a blue Berge-G 1 or a red Berge-G 2 subhypergraph. We show that for every k ≥ 2, there exists some c k such that for any finite graphs G

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Section: Introductionmentioning

confidence: 98%

On the cover Ramsey number of Berge hypergraphs

Lü¹,

Wang²

2019

Preprint

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…The case when k = r remained open. Both papers [9,4] conjectured the maximum number of edges to be bounded by max (n−1)(r−1) r , n − (r − 1) (See Figure 1). Theorem 5 (Füredi, Kostochka and Luo [5,6]).…”

Section: Introductionmentioning

confidence: 99%

The Structure of Hypergraphs without long Berge cycles

Győri¹,

Lemons²,

Salia³

et al. 2018

Preprint

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“…These bounds were improved by Füredi and Özkahya [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].…”

Section: Introductionmentioning

confidence: 99%