Mono-multi bipartite Ramsey numbers, designs, and matrices

Abstract

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, , K r,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(K 1, , K 2,2 ) = 3 − 2 and that the smallest n for which any edge coloring of K ,n contains either a monochromatic K 1, or a multicolored K 2,2 is 2 .

Rainbow generalizations of Ramsey theory - a dynamic survey

“…This bound implies the following exact result for small paths. Also, in [26], the following is shown.…”

Constrained Ramsey Numbers

“…The constrained Ramsey number has been studied by many researchers [1,3,4,7,8,10,13,14,18], and the bipartite case in [2]. In the special case when H = K 1,k+1 is a star with k + 1 edges, colourings with no rainbow H have the property that every vertex is incident to edges of at most k different colours, and such colourings are called k-local.…”

Constrained Ramsey Numbers

“…The constrained Ramsey number has been studied by many researchers [1,3,4,7,8,10,13,14,18], and the bipartite case in [2]. In the special case when H = K 1,k+1 is a star with k + 1 edges, colorings with no rainbow H have the property that every vertex is incident to edges of at most k different colors, and such colorings are called k-local.…”

“…This substantially improves the previous bounds for most values of s and t.1 has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T . It is an immediate consequence of the Canonical Ramsey Theorem that this number exists if and only if S is a star or T is acyclic, because stars are the only graphs that admit a simultaneously lexical and monochromatic coloring, and forests are the only graphs that admit a simultaneously lexical and rainbow coloring.The constrained Ramsey number has been studied by many researchers [1,3,4,7,8,10,13,14,18], and the bipartite case in [2]. In the special case when H = K 1,k+1 is a star with k + 1 edges, colorings with no rainbow H have the property that every vertex is incident to edges of at most k different colors, and such colorings are called k-local.…”