On connected components with many edges

Abstract: We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then $H$ contains a connected component $H'$ satisfying $|E(H')||E(G)|\geq |E(H)|^2$. 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. Moreover, for general $k$, we show that every $k$-coloring of the edges of $K_n$ contains a monochromatic connected subgraph with at least $\frac{1}{k^2-k+\frac{5}{4}}\binom{n}{2}$ edges.