For a graph G and an integer t we let mcct(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that mcc2(G)=O(n2/3) 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(n2/(t+1)). On the other hand, we have examples of graphs G with no Kt+3 minor and with mcct(G)=Ω(n(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 ε>0 there exists cε>0 and n-vertex graphs of maximum degree 7, average degree at most 6+ε for all subgraphs, and with mcc2(G)⩾cεn. 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).