The Ramsey Number of Loose Paths in 3-Uniform Hypergraphs
نویسندگان
چکیده
منابع مشابه
The Ramsey Number of Loose Paths in 3-Uniform Hypergraphs
Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of 3-uniform loose paths when one of the paths is significantly larger than the other: for every n > ⌊ 5m 4 ⌋ , we show that R(P n,P m) = 2n + ⌊m + 1 2 ⌋ .
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Asymptotic values of hypergraph Ramsey numbers for loose cycles (and paths) were determined recently. Here we determine some of them exactly, for example the 2-color hypergraph Ramsey number of a k-uniform loose 3-cycle or 4-cycle: R(Ck 3 , Ck 3 ) = 3k − 2 and R(Ck 4 , Ck 4 ) = 4k − 3 (for k > 3). For more than 3 colors we could prove only that R(C3 3 , C3 3 , C3 3) = 8. Nevertheless, the r-col...
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The values of hypergraph 2-color Ramsey numbers for loose cycles and paths have already been determined. The only known value for more than 2 colors is R(C 3 ; 3) = 8, where C 3 3 is a 3-uniform loose cycle of length 3. Here we determine that R(P 3 3 ; 3) = 9, where P 3 3 is a 3-uniform loose path of length 3. Our proof relies on the determination of the Turán number ex3(9;P 3 3 ). We also find...
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A conjecture of Gyárfás and Sárközy says that in every 2-coloring of the edges of the complete k-uniform hypergraph Kk n, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most k − 2 vertices. Recently, the authors affirmed the conjecture. In the note we show that for every 2-coloring of Kk n, one can find two monochromatic paths of distinct col...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2013
ISSN: 1077-8926
DOI: 10.37236/2725