The m-Bipartite Ramsey Number of the K{2,2} Versus K{6,6}

Abstract

For the given bipartite graphs G 1 , . . . , G n , the bipartite Ramsey number BR ( G 1 , . . . , G n ) is the least positive integer b such that any complete bipartite graph K b,b having edges coloured with 1 , 2 , . . . , n , contains a copy of some G i ( 1 ≤ i ≤ n ), where all the edges of G i have colour i . For the given bipartite graphs G 1 , . . . , G n and a positive integer m , the m -bipartite Ramsey number BR m ( G 1 , . . . , G n ) is defined as the least positive integer b ( b ≥ m ) such that any complete bipartite graph K m,b having edges coloured with 1 , 2 , . . . , n , contains a copy of some G i ( 1 ≤ i ≤ n ), where all the edges of G i have colour i . The values of BR m ( G 1 , G 2 ) (for each m ), BR m ( K 3 , 3 , K 3 , 3 ) and BR m ( K 2 , 2 , K 5 , 5 ) (for particular values of m ) have already been determined in several articles, where G 1 = K 2 , 2 and G 2 ∈ { K 3 , 3 , K 4 , 4 } . In this article, the value of BR m ( K 2 , 2 , K 6 , 6 ) is computed for each m ∈ { 2 , 3 , . . . , 8 } .