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| Paper IPM / M / 18009 | ||||||||||||||||||
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For given k-uniform hypergraphs G and H, the Ramsey number R(G,H) is the smallest positive integer N such that in every red-blue coloring of the edges of the complete k-uniform hypergraph on n vertices there is either a red copy of G or a blue copy of H. In this paper, results are given which permit the R(mG,nH) to be evaluated exactly when m or n is large and G is a
k-uniform hypergraph with the maximum independent set that intersects each edge 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, 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 when m or n is large and G, H are either loose paths, loose cycles, tight paths or stars. Moreover, for given k-uniform hypergraphs G and H and positive integers m,n, some bounds are given for R(mG,nH) which enable us to compute R(mG,nH) when m≥n≥1 and
G,H are either 3-uniform loose path P3r or loose cycle C3r: We shall show
that for every m≥n≥1 and r≥s, R(mC3r,nC3s)=2rm+⌊s+12⌋n−1, and R(mP3r,nP3s)=(2r+1)m+⌊s+12⌋n−1.
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