Large Monochromatic Components in Almost Complete Graphs and Bipartite Graphs

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