  Given an $r$-edge-coloring of the complete graph $K_n$, 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 $K_n$ contains a monochromatic component with at least
$\frac{1}{12}\binom{n}{2}$ edges, where the constant $\frac{1}{12}$ is optimal
only when the coloring matches a certain construction of Gy\'arf\'as.
