  For every $n\in\mathbb{N}$ and $k\geq2$, it is known that every
$k$-edge-colouring of the complete graph on $n$ vertices contains a
monochromatic connected component of order at least $\frac{n}{k-1}$. For
$k\geq3$, it is known that the complete graph can be replaced by a graph $G$
with $\delta(G)\geq(1-\varepsilon_k)n$ for some constant $\varepsilon_k$. In
this paper, we show that the maximum possible value of $\varepsilon_3$ is
$\frac16$. This disproves a conjecture of Gy\'{a}rfas and S\'{a}rk\"{o}zy.
