Multicolor bipartite Ramsey numbers of Kt,s and large Kn,n

For any bipartite graph H, let us denote the bipartite Ramsey number brk(H;Kn,n) to be the minimum integer N such that any edge-coloring of the complete bipartite graph KN,N by k+1 colors contains a monochromatic copy of H in some color i for 1≤i≤k, or a monochromatic copy of Kn,n in the last color. In this note, it is shown that for any fixed integers t≥2 and s≥(t−1)!+1, there exists a constant c=c(t)>0 such that br2(Kt,s;Kn,n)≥c(nloglognlog2n)t for sufficiently large n; and for k≥3, brk(Kt,s;Kn,n)=Θ(ntlogtn).