Gyárfás and Sárközy's minimum-degree conjecture for monochromatic components

Fix $k\geq 3$. Let $G$ be a graph with $n$ vertices and minimum degree satisfying

$$\delta(G)\geq\left(1-\frac{k-1}{k^2}\right)n.$$

If the edges of $G$ are $k$-coloured, meaning that each edge receives one of $k$ colours, then Gyárfás and Sárközy's conjecture. there exists a monochromatic component of order at least

$$\frac{n}{k-1}.$$

This conjecture asks how far the complete-graph result for $k$-edge-colourings extends to graphs with large but not necessarily complete minimum degree. The source notes that Gyárfás and Sárközy had already proved the existence of some constant $\varepsilon_k>0$ for which minimum degree at least $(1-\varepsilon_k)n$ suffices; the stated threshold and conclusion remain the conjectured sharp form.