Graph colouring with no large monochromatic components

For a graph G and an integer t we let mcct(G) be the smallest m such that there exists a colouring of the vertices of G by t colours with no monochromatic connected subgraph having more than m vertices. Let F be any non-trivial minor-closed family of graphs. We show that mcc2(G) = O(n^2/3) for any n-vertex graph G ∈ F. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F, and every fixed t we show that mcct(G)=O(n^2/(t+1)). On the other hand, we have examples of graphs G with no Kt+3 minor and with mcct(G)=On^2/(2t-1)). It is also interesting to consider graphs of bounded degrees. Haxell, Szabó and Tardos proved mcc2(G) ≤ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc2(G)=ω(n), and more sharply, for every > 0 there exists c > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + for all subgraphs, and with mcc2(G) ≤ cn. For 6-regular graphs it is known only that the maximum order of magnitude of mcc2 is between n and n. We also offer a Ramsey-theoretic perspective of the quantity mcct(G).