Ramsey Numbers for Nontrivial Berge Cycles

Nie, Jiaxi; Verstraëte, Jacques

doi:10.1137/21m1396770

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…The conjecture is true for k = 3 due to (1). It is shown in [25] that R(t, C 3 4 ) ≤ t 4/3+o (1) , and Méroueh [21] showed R(t, C r k ) = O(t 1+1/⌊(k+1)/2⌋ ) for k ≥ 3, improving earlier results of Collier-Cartaino, Graber and Jiang [9]. Conjecture I motivates our current study of nontrivial Berge k-cycles.…”

Section: Introductionsupporting

confidence: 76%

Ramsey Numbers for Non-trivial Berge Cycles

Nie¹,

Verstraëte²

2021

Preprint

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In this paper, we consider an extension of cycle-complete graph Ramsey numbers to Berge cycles in hypergraphs: for k ≥ 2, a non-trivial Berge k-cycle is a family of sets e 1 , e 2 , . . . , e k such that e 1 ∩e 2 , e 2 ∩e 3 , . . . , e k ∩e 1 has a system of distinct representatives and e 1 ∩ e 2 ∩ • • • ∩ e k = ∅. In the case that all the sets e i have size three, let B k denotes the family of all non-trivial Berge k-cycles. The Ramsey numbers R(t, B k ) denote the minimum n such that every n-vertex 3-uniform hypergraph contains either a non-trivial Berge k-cycle or an independent set of size t. We proveand moreover, we show that if a conjecture of Erdős and Simonovits [11] on girth in graphs is true, then this is tight up to a factor t o(1) as t → ∞.

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Section: Introductionsupporting

confidence: 76%

Ramsey Numbers for Non-trivial Berge Cycles

Nie¹,

Verstraëte²

2021

Preprint

Self Cite

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“…Since the introduction of the Berge-Turán problem, a long list of papers have been written about it, far too many to cite here, and we recommend [11] for a partial history. More recently Ramsey problems have also been considered, see for example [3,4,5,9,10,13,14,19,21,22,23]. The two problems are related as any coloring avoiding monochromatic B(G) must have that every color class contains at most ex r (n, B(G)) edges.…”

Section: Introductionmentioning

confidence: 99%

Multicolor Ramsey Numbers for Berge cycles

DeStefano

Mahon

Simutis

et al. 2022

Electron. J. Combin.

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In this paper, for small uniformities, we determine the order of magnitude of the multicolor Ramsey numbers for Berge cycles of length $4$, $5$, $6$, $7$, $10$, or $11$. Our result follows from a more general setup which can be applied to other hypergraph Ramsey problems. Using this, we additionally determine the order of magnitude of the multicolor Ramsey number for Berge-$K_{a,b}$ for certain $a$, $b$, and uniformities.

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Ramsey Numbers for Nontrivial Berge Cycles

Cited by 2 publications

References 19 publications

Ramsey Numbers for Non-trivial Berge Cycles

Ramsey Numbers for Non-trivial Berge Cycles

Multicolor Ramsey Numbers for Berge cycles

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