Ramsey Numbers for Multiple Copies of Hypergraphs

For given k-uniform hypergraphs \({\mathcal {G}}\) and \({\mathcal {H}}\), the Ramsey number \(R({\mathcal {G}},{\mathcal {H}})\) 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 \({\mathcal {G}}\) or a blue copy of \({\mathcal {H}}\). In this paper, results are given which permit the \(R(m{\mathcal {G}},n{\mathcal {H}})\) to be evaluated exactly when m or n is large and \({\mathcal {G}}\) is a k-uniform hypergraph with the maximum independent set that intersects each edge in \(k-1\) vertices and \({\mathcal {H}}\) 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 \({\mathcal {G}}\) and \({\mathcal {H}}\), 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 \({\mathcal {G}}\) and linear hypergraphs for \({\mathcal {H}}\). As an application, \(R(m{\mathcal {G}},n{\mathcal {H}})\) is determined when m or n is large and \({\mathcal {G}}\), \({\mathcal {H}}\) are either loose paths, loose cycles, tight paths or stars. Moreover, for given k-uniform hypergraphs \({\mathcal {G}}\) and \({\mathcal {H}}\) and positive integers m, n, some bounds are given for \(R(m{\mathcal {G}},n{\mathcal {H}})\) which enable us to compute \(R(m{\mathcal {G}},n{\mathcal {H}})\) when \(m\ge n\ge 1\) and \({\mathcal {G}}, {\mathcal {H}}\) are either 3-uniform loose path \({\mathcal {P}}_r^3\) or loose cycle \({\mathcal {C}}_r^3\): We shall show that for every \(m\ge n\ge 1\) and \(r\ge s\), \(R(m{\mathcal {C}}_r^3,n{\mathcal {C}}_s^3)=2rm+\Big \lfloor \frac{s+1}{2}\Big \rfloor n-1,\) and \(R(m{\mathcal {P}}_r^3,n{\mathcal {P}}_s^3)=(2r+1)m+\Big \lfloor \frac{s+1}{2}\Big \rfloor n-1.\)

The authors would like to thank the anonymous referees for their valuable comments that improved the quality of the manuscript. The research of the second author was in part supported by a grant from IPM No. 1403050321.

The authors have not disclosed any funding.

Authors and Affiliations

1. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran

   Gholam Reza Omidi

2. Department of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran

   Ghaffar Raeisi

3. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, 19395-5746, Iran

   Ghaffar Raeisi

Corresponding author

Correspondence to Ghaffar Raeisi.

Conflict of interest

The authors declare that they have no Conflict of interest.

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