Mono-multi bipartite Ramsey numbers, designs, and matrices

Eroh and Oellermann defined BRR ( G 1 , G 2 ) as the smallest N such that any edge coloring of the complete bipartite graph K N , N contains either a monochromatic G 1 or a multicolored G 2 . We restate the problem of determining BRR ( K 1 , λ , K r , s ) in matrix form and prove estimates and exact values for several choices of the parameters. Our general bound uses Fürediʹs 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 ( K 1 , λ , K 2 , 2 ) = 3 λ - 2 and that the smallest n for which any edge coloring of K λ , n contains either a monochromatic K 1 , λ or a multicolored K 2 , 2 is λ 2 .