A note on Ramsey numbers for Berge-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" id="d1e80" altimg="si33.gif"><mml:mi>G</mml:mi></mml:math> hypergraphs

Axenovich, Maria; Gyárfás, András

doi:10.1016/j.disc.2019.01.011

Cited by 18 publications

(23 citation statements)

References 15 publications

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“…The upper bound of Theorem 3 gives the upper bound in Theorem 2. In fact, it also matches the corresponding asymptotic lower bound 2 n−1 |E(G)|−1 (1 − o(1)) in [1] when G is a cycle or path, and the asymptotic lower bound of Proposition 2, implying (1)).…”

Section: Introduction and Resultssupporting

confidence: 80%

“…In case of G = C 4 we improve the upper bound of Proposition 1 by a constant factor and slightly improve the lower bound 2 n−1 3 (1 − o(1)) from [1]. Theorem 2.…”

Section: Introduction and Resultsmentioning

confidence: 76%

“…For the case of the triangle, the upper bound of Proposition 1 is tight. For odd n ≥ 7 this was shown with a simple proof in [1]. Somewhat surprisingly, this remains true for the even n case as well (but not for n = 5).…”

Section: Introduction and Resultsmentioning

confidence: 88%

“…An exception is the Finite Unions Theorem of Folkman, Rado, Sanders [6]. A more recent research in this direction is by Axenovich and Gyárfás [1], where Ramsey numbers of Berge-G hypergraphs were studied for several graphs G in colorings of P(n). Ramsey numbers of Berge-G hypergraphs in the uniform case have been investigated also in [5,9].…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Since A and [n − 1] \ A define the same part, we have 2 n−2 parts (each of size four). This partition was used in [1] to show that f (n, C 3 ) ≤ 2 n−2 . Observing that…”

Section: Introduction and Resultsmentioning

confidence: 99%

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Partitioning the Power Set of $[n]$ into $C_k$-free parts

Blaisdell

Gyárfás

Krueger

et al. 2019

Electron. J. Combin.

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We show that for n ≥ 3, n = 5, in any partition of P(n), the set of all subsets of [n] = {1, 2, . . . , n}, into 2 n−2 − 1 parts, some part must contain a triangle -three different subsets A, B, C ⊆ [n] such that A ∩ B, A ∩ C, and B ∩ C have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into 2 n−2 triangle-free parts. We also address a more general Ramsey-type problem: for a given graph G, find (estimate) f (n, G), the smallest number of colors needed for a coloring of P(n), such that no color class contains a Berge-G subhypergraph. We give an upper bound for f (n, G) for any connected graph G which is asymptotically sharp (for fixed k) when G = C k , P k , S k , a cycle, path, or star with k edges. Additional bounds are given for G = C 4 and G = S 3 .

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Section: Introduction and Resultssupporting

confidence: 80%

“…In case of G = C 4 we improve the upper bound of Proposition 1 by a constant factor and slightly improve the lower bound 2 n−1 3 (1 − o(1)) from [1]. Theorem 2.…”

Section: Introduction and Resultsmentioning

confidence: 76%

Section: Introduction and Resultsmentioning

confidence: 88%

Section: Introduction and Resultsmentioning

confidence: 99%

“…Since A and [n − 1] \ A define the same part, we have 2 n−2 parts (each of size four). This partition was used in [1] to show that f (n, C 3 ) ≤ 2 n−2 . Observing that…”

Section: Introduction and Resultsmentioning

confidence: 99%

See 3 more Smart Citations