Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree

For every $n\in\mathbb{N}$ and $k\geqslant2$, it is known that $k$-edge-colouring of the complete graph on $n$ vertices contains a monochromatic connected component order at least $\frac{n}{k-1}$. $k\geqslant3$, can be replaced by $G$ with $\delta(G)\geqslant(1-\varepsilon_k)n$ for some constant $\varepsilon_k$. In this paper, we show maximum possible value $\varepsilon_3$ $\frac16$. This disproves conjecture Gyárfas Sárközy.