Monochromatic diameter-2 components in edge colorings of the complete graph

Abstract

Gyárfás conjectured that in every $r$-edge-coloring of the complete graph $K_n$ there is a monochromatic component on at least $n∕\left(r-1\right)$ vertices which has diameter at most 3. We show that for $r=3,4,5$ and $6$ a diameter of 3 is best possible in this conjecture, constructing colorings where every monochromatic diameter-2 subgraph has strictly less than $n∕\left(r-1\right)$ vertices.