General lemmas for Berge–Turán hypergraph problems

Gerbner, Dániel; Methuku, Abhishek; Palmer, Cory

doi:10.1016/j.ejc.2020.103082

Cited by 38 publications

(33 citation statements)

References 43 publications

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“…The non-uniform and multi-hypergraph case was addressed in that paper too. Other results on extremal numbers for Berge hypergraphs were provided by Gerbner and Palmer [5], Gerbner, Methuku, and Vizer, [3], Gerbner, Methuku, and Palmer, [4], as well as by Palmer, Tait, Timmons, and Wagner [12], and by Grósz, Methuku, and Tompkins [7].…”

Section: Introductionmentioning

confidence: 99%

A note on Ramsey numbers for Berge-G hypergraphs

Axenovich

Gyárfás

2019

Discrete Mathematics

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Section: Introductionmentioning

confidence: 99%

A note on Ramsey numbers for Berge-G hypergraphs

Axenovich

Gyárfás

2019

Discrete Mathematics

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“…Füredi andÖzkahya [21] obtained better constant factors (depending on k). Further improvements were obtained for even k by Gerbner, Methuku and Vizer [24], also by Gerbner, Methuku and Palmer [23], and for odd k by Gerbner [22]. For s ≥ 4, Győri and Lemons [26] showed that ex(n, s, BC 2k+1 ) ≤ O s (k s−2 ) · ex(n, 3, BC 2k+1 ) and ex(n, s, BC 2k ) ≤ O s (k s−1 ) · ex(n, C 2k ), i.e., ex(n, s, BC k ) = O(n 1+1/⌊k/2⌋ ).…”

Section: Preliminariesmentioning

confidence: 88%

Anti-Ramsey Numbers of Paths and Cycles in Hypergraphs

Gu¹,

Li²,

Shi³

2020

SIAM J. Discrete Math.

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The anti-Ramsey problem was introduced by Erdős, Simonovits and Sós in 1970s. The anti-Ramsey number of a hypergraph H, ar(n, s, H), is the smallest integer c such that in any coloring of the edges of the s-uniform complete hypergraph on n vertices with exactly c colors, there is a copy of H whose edges have distinct colors. In this paper, we determine the anti-Ramsey numbers of linear paths and loose paths in hypergraphs for sufficiently large n, and give bounds for the anti-Ramsey numbers of Berge paths. Similar exact anti-Ramsey numbers are obtained for linear/loose cycles, and bounds are obtained for Berge cycles. Our main tools are path extension technique and stability results on hypergraph Turán problems of paths and cycles.

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“…Extremal problems for Berge hypergraphs have attracted the attention of many researchers, see e.g. [6,7,9,10,19,20,21,24].…”

Section: Introductionmentioning

confidence: 99%

Ramsey Problems for Berge Hypergraphs

Gerbner¹,

Methuku²,

Omidi³

et al. 2020

SIAM J. Discrete Math.

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For a graph G, a hypergraph H is a Berge copy of G (or a Berge-G in short), if there is a bijection f : E(G) → E(H) such that for each e ∈ E(G) we have e ⊆ f (e). We denote the family of r-uniform hypergraphs that are Berge copies of G by B r G.For families of r-uniform hypergraphs H and H ′ , we denote by R(H, H ′ ) the smallest number n such that in any red-blue coloring of the (hyper)edges of K r n (the complete r-uniform hypergraph on n vertices) there is a monochromatic blue copy of a hypergraph in H or a monochromatic red copy of a hypergraph in H ′ . R c (H) denotes the smallest number n such that in any coloring of the hyperedges of K r n with c colors, there is a monochromatic copy of a hypergraph in H.In this paper we initiate the general study of the Ramsey problem for Berge hypergraphs, and show that if r > 2c, then R c (B r K n ) = n. In the case r = 2c, we show that R c (B r K n ) = n + 1, and if G is a non-complete graph on n vertices, then R c (B r G) = n, assuming n is large enough. In the case r < 2c we also obtain bounds on R c (B r K n ). Moreover, we also determine the exact value of R(B 3 T 1 , B 3 T 2 ) for every pair of trees T

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General lemmas for Berge–Turán hypergraph problems

Cited by 38 publications

References 43 publications

A note on Ramsey numbers for Berge-G hypergraphs

A note on Ramsey numbers for Berge-G hypergraphs

Anti-Ramsey Numbers of Paths and Cycles in Hypergraphs

Ramsey Problems for Berge Hypergraphs

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