Bipartite rainbow Ramsey numbers

Mono-multi bipartite Ramsey numbers, designs, and matrices

“…Matrices obtained by row or column permutations or by transformations on the values of the entries preserving equality are considered to be the same. We shall prove (Theorem 5) that n (2, 2) = 3 − 2, implying that the lower bound in [8] is the true value of BRR(K 1, , K 2,2 ). For the asymmetric version, we show (Theorem 7) that n (2, 2; ) = 2 and the extremal matrix is unique.…”

“…It seems that the variant when either monochromatic or multicolored configurations are sought-perhaps one may call it monomulti Ramsey numbers-was mentioned first in [1], although the concept (for arithmetic progressions) already appeared in [5]. Without completeness, we point out some recent references on the subject: [8,11,16].…”

“…Eroh and Oellermann [8] considered the mono-multi Ramsey numbers for bipartite graphs: they defined BRR(G 1 , G 2 ) as the smallest N such that any edge coloring of K N,N contains either a monochromatic G 1 or a multicolored G 2 . They proved that this number exists if and only if G 1 is a star or G 2 is a star forest.…”

“…They proved that this number exists if and only if G 1 is a star or G 2 is a star forest. Among many results they proved that BRR(K 1, , K r,s ) = O( ) for fixed r, s (in [8,Theorem 2]). In particular, for r = s = 2 (i.e., when G 2 is a 4-cycle) they showed that 3 − 2 BRR(K 1, , K 2,2 ) 6 − 8 (see [8, p. 71]).…”

Bipartite Ramsey numbers involving stars, stripes and trees

“…Exact solutions were given for simpler cases of the problem such as path-path bipartite Ramsey numbers [9,11], star-star bipartite Ramsey numbers [17], star-path bipartite Ramsey numbers [13], K 2,2 -K 1,n and K 2,2 -K 2,n bipartite Ramsey numbers [2], C 2m -K 2,2 bipartite Ramsey numbers [21] and bipartite Ramsey numbers for multiple copies of K 2,2 [14]. Some variations of the bipartite case such as multicolour problems [3,5] and rainbow colouring problems [7] have been also studied.…”