On multicolor Ramsey number of paths versus cycles

zbMATH Open1217.05093arXiv1303.0474MaRDI QIDQ625389

Ghaffar Raeisi, Gholam Reza Omidi

Publication date: 17 February 2011

Published in: The Electronic Journal of Combinatorics (Search for Journal in Brave)

Abstract: In this paper, for sufficiently large $n$ we determine the Ramsey number $R(G,nH)$ where $G$ is a $k$-uniform hypergraph with the maximum independent set that intersects each of the edges in $k-1$ vertices and $H$ is a $k$-uniform hypergraph with a vertex so that the hypergraph induced by the edges containing this vertex is a star. There are several examples for such $G$ and $H$, among them are any disjoint union of $k$-uniform hypergraphs involving loose paths, loose cycles, tight paths, tight cycles with a multiple of $k$ edges, stars, Kneser hypergraphs and complete $k$-uniform $k$-partite hypergraphs for $G$ and linear hypergraphs for $H$. As an application, $R(mG,nH)$ is determined where $m$ or $n$ is large and $G$ and $H$ are either loose paths, loose cycles, tight paths, or stars. Also, $R(G,nH)$ is determined when $G$ is a bipartite graph with a matching saturating one of its color classes and $H$ is an arbitrary graph for sufficiently large $n$. Moreover, some bounds are given for $R(mG,nH)$ which allow us to determine this Ramsey number when $mgeqn$ and $G$ and $H$, $(|V(G)|geq|V(H)|)$, are 3-uniform loose paths or cycles, $k$-uniform loose paths or cycles with at most 4 edges and $k$-uniform stars with 3 edges.

Full work available at URL: https://arxiv.org/abs/1303.0474

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zbMATH Keywords

pathscyclesRamsey nubmer

Mathematics Subject Classification ID

Extremal problems in graph theory (05C35) Coloring of graphs and hypergraphs (05C15) Generalized Ramsey theory (05C55)

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Note on the multicolour size-Ramsey number for paths ⋮ Some multi-color Ramsey numbers on stars versus path, cycle or wheel ⋮ On three-color Ramsey number of paths ⋮ Three-color Ramsey number of an odd cycle versus bipartite graphs with small bandwidth ⋮ On some three color Ramsey numbers for paths, cycles, stripes and stars ⋮ Unnamed Item

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