Conlon–Luo–Tyomkyn conjecture for monochromatic components in complete graphs

Prove that for every pair of positive integers n and r with r≥3, every r-coloring of the edges of the complete graph K_n contains a monochromatic connected component with at least \(\frac{1}{r(r-1)}\binom{n}{2}\) edges.

Background

The paper studies the maximum number of edges that a monochromatic connected component must contain in an r-edge-coloring of a complete graph. Gyárfás's construction gives, for infinitely many values of r, colorings in which every monochromatic component has at most $(1/(r(r−1))+o(1))\binom{n}{2}$ edges, so the conjectured lower bound would be asymptotically tight.

The cases r=3 and r=4 had been established, while the cited general result provided only a lower bound of $1/(r^2-r+5/4)\binom{n}{2}$. The paper proves analogues of this weaker bound for sparse random graphs and high-density graphs, but does not resolve the stated complete-graph conjecture for all r.