Monochromatic components in edge-coloured graphs with large minimum degree (Q2223462)

Summary: For every \(n\in\mathbb{N}\) and \(k\geqslant 2\), 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 \(k\geqslant 3\), it is known that the complete graph can be replaced by a graph \(G\) with \(\delta(G)\geqslant(1-\varepsilon_k)n\) 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 \textit{A. Gyárfás} and \textit{G. Sárközy} [ibid. 24, No. 3, Research Paper P3.54, 14 p. (2017; Zbl 1369.05076)].