Large monochromatic components in almost complete graphs and bipartite graphs

Abstract

Gyárfas proved that every coloring of the edges of $K_n$ with $t+1$ colors contains a monochromatic connected component of size at least $n/t$. Later, Gyárfás and Sárközy asked for which values of $\gamma =\gamma (t)$ does the following strengthening for almost complete graphs hold: if $G$ is an $n$-vertex graph with minimum degree at least $(1-\gamma )n$, then every $(t+1)$-edge coloring of $G$ contains a monochromatic component of size at least $n/t$. We show $\gamma =1/(6t^3)$ suffices, improving a result of DeBiasio, Krueger, and Sárközy.