Tiling with monochromatic bipartite graphs of bounded maximum degree

We prove that for any r is an element of N, there exists a constant C-r such that the following is true. Let F={F1,F2,...}be an infinite sequence of bipartite graphs such that |V(F-i)|=i| and Delta(F-i) <= Delta hold for all i. Then, in any r-edge-coloured complete graph K-n, there is a collection of at most exp(C-r Delta) monochromatic subgraphs, each of which is isomorphic to an element of F, whose vertex sets partition V(K-n). This proves a conjecture of Corsten and Mendonca in a strong form and generalises results on the multi-colour Ramsey numbers of bounded-degree bipartite graphs. It also settles the bipartite case of a general conjecture of Grinshpun and Sarkozy.