On connected components with many edges

We prove that if H is a subgraph of a complete multipartite graph G, then H contains a connected component H ′ satisfying |E(H ′)||E(G)| ≥ |E(H)|. We use this to prove that every three-coloring of the edges of a complete graph contains a monochromatic connected subgraph with at least 1/6 of the edges. We further show that such a coloring has a monochromatic circuit with a fraction 1/6− o(1) of the edges. This verifies a conjecture of Conlon and Tyomkyn.