Mono-multi bipartite Ramsey numbers, designs, and matrices

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Abstract

Eroh and Oellermann defined BRR(G1,G2) as the smallest N such that any edge coloring of the complete bipartite graph KN,N contains either a monochromatic G1 or a multicolored G2. We restate the problem of determining BRR(K1,λ,Kr,s) in matrix form and prove estimates and exact values for several choices of the parameters. Our general bound uses Füredi's result on fractional matchings of uniform hypergraphs and we show that it is sharp if certain block designs exist. We obtain two sharp results for the case r=s=2: we prove BRR(K1,λ,K2,2)=3λ-2 and that the smallest n for which any edge coloring of Kλ,n contains either a monochromatic K1,λ or a multicolored K2,2 is λ2.

Keywords

Ramsey theory
Mono- and multicolored bipartite graphs
Rainbow matrix
Extremal configurations
3-design

Cited by (0)

1

Research partially supported by NSF Grant EIA-0130352.

2

Thanks for the hospitality of the Institute of Combinatorics at the University of Memphis while on leave from the Computer and Automation Research Institute of the Hungarian Academy of Sciences.

3

On leave from the Computer and Automation Research Institute of the Hungarian Academy of Sciences.