LAUSR

Given bipartite graphs G and H, the bipartite rainbow Ramsey number BRR(G; H) is the minimum integer N such that any edge-coloring of $$K_{N,N}$$KN,N with any number of colors contains either a monochromatic copy of G or a rainbow copy of H. It is known that BRR(G; H) exists if and only if G is a star or H is a forest consisting of stars. For fixed $$t\ge 3$$t≥3, $$s\ge (t-1)!+1$$s≥(t-1)!+1 and large n, we shall show that $$BRR(K_{t,s};K_{1,n})=\varTheta (n^t)$$BRR(Kt,s;K1,n)=Θ(nt) and $$BRR(K_{1,n};K_{t,t})=\varTheta (n)$$BRR(K1,n;Kt,t)=Θ(n). We also improve the known bounds for $$ BRR(C_{2m};K_{1,n})$$BRR(C2m;K1,n), $$BRR(K_{1,n};C_{2m})$$BRR(K1,n;C2m), $$BRR(B_{s,t};K_{1,n})$$BRR(Bs,t;K1,n) and $$BRR(K_{1,n};B_{s,t})$$BRR(K1,n;Bs,t), where $$B_{s,t}$$Bs,t is a broom consisting of $$s+t$$s+t edges obtained by identifying the center of star $$K_{1,s}$$K1,s with an end-vertex of a path $$P_{1+t}$$P1+t. Particularly, we have $$BRR(C_{2m};K_{1,n})\ge (1-o(1))n^{m/(m-1)}$$BRR(C2m;K1,n)≥(1-o(1))nm/(m-1) for $$m=2,3,5$$m=2,3,5 and large n.