Monochromatic components with many edges

Given an r-edge-coloring of the complete graph Kam,, what is the largest number of edges in a monochromatic connected component? This natural question has only recently received the attention it deserves, with work by two disjoint subsets of the authors resolving it for the first two special cases, when r = 2 or 3. Here we introduce a general framework for studying this problem and apply it to fully resolve the r = 4 case, showing that any 4-edge-coloring of Kam, (am, contains a monochromatic component with at least 1 ) edges, 12 2 where the constant 112 is optimal only when the coloring matches a certain construction of Gyarfas.