The Ramsey number of generalized loose paths in 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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Let H = (V, E) be an r-uniform hypergraph. For each 1 ≤ s ≤ r − 1, an spath P n in H of length n is a sequence of distinct vertices v1, v2, . . . , vs+n(r−s) such that {v1+i(r−s), . . . , vs+(i+1)(r−s)} ∈ E(H) for each 0 ≤ i ≤ n − 1. Recently, the Ramsey number of 1-paths and 1-cycles in uniform hypergraphs attracted a lot of attention. The asymptotic value of R(P n ,P 3,1 n ) was first determi...
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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 2-color Ramsey number R(C3 n, C3 n) of a 3-uniform loose cycle Cn is asymptotic to 5n/4 as has been recently proved by Haxell, Luczak, Peng, Rödl, Ruciński, Simonovits and Skokan. Here we extend their result to the r-uniform case by showing that the corresponding Ramsey number is asymptotic to (2r−1)n 2r−2 . Partly as a tool, partly as a subject of its own, we also prove that for r ≥ 2, R(k...
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The size-Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2-edge-coloring of H yields a monochromatic copy of G. Size-Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size-Ramsey numbers for k-uniform hypergraphs. Analogous to the graph case, we cons...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2016
ISSN: 0012-365X
DOI: 10.1016/j.disc.2015.09.023