Three-colour bipartite Ramsey number for graphs with small bandwidth

We estimate the $3$-colour bipartite Ramsey number for balanced bipartite graphs $H$ with small bandwidth and bounded maximum degree. More precisely, we show that the minimum value of $N$ such that in any $3$-edge colouring of $K_{N,N}$ there is a monochromatic copy of $H$ is at most $(3\mathrm{/}2+o(1))\mathrm{\mid }V(H)\mathrm{\mid }$. In particular, we determine asymptotically the $3$-colour bipartite Ramsey number for balanced grid graphs.