Local colourings and monochromatic partitions in complete bipartite graphs

We show that for any 2-local colouring of the edges of the balanced complete bipartite graph Kn,nKn,n, its vertices can be covered with at most 3 disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite rr-locally coloured graph with O(r2)O(r2) disjoint monochromatic cycles. We also determine the 2-local bipartite Ramsey number of a path almost exactly: Every 2-local colouring of the edges of Kn,nKn,n contains a monochromatic path on nn vertices.