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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.

Large Monochromatic Components in Almost Complete Graphs and Bipartite Graphs