Ramsey numbers of connected clique matchings

We determine the Ramsey number of a connected clique matching. That is, we show that if GG is a 22-edge-coloured complete graph on (r2−r−1)n−r+1(r2−r−1)n−r+1 vertices, then there is a monochromatic connected subgraph containing nn disjoint copies of KrKr, and that this number of vertices cannot be reduced.