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For simple bipartite graphs G 1 , G 2 , G 3 , the three-colour bipartite graph Ramsey number R b ( G 1 , G 2 , G 3 ) is defined as the least positive integer n such that any 3-edge-colouring of K n,n assures a monochromatic copy of G i in the i th colour for some i , i ∈ {1,2,3}. In this paper, we consider the three-colour bipartite Ramsey number R b ( G 1 , G 2 , P 3 ). Exact values are determined when G 1  = G 2  = C 4  and when ( G 1 , G 2 ) = (a bistar, a bistar). For integers m,n ≥ 2, a recursive upper bound, R b ( K m,m , K n , n , P 3 ) ≤ R b ( K m -1, m -1 , K n , n , P 3 ) +  R b ( K m,m , K n -1, n -1 , P 3 ) + 3, is given. When G 1  and G 2  are even cycles, a lower bound is provided. In addition to these results, we have obtained the relations: R ( G , K 1, n ) ≤  R b ( G , K 1, n +1 ) and R ( G , H ) ≤ R b ( G , H , P 3 ).

Three-colour bipartite Ramsey number $R_b(G_1,G_2,P_3)$