Vertex covering with monochromatic pieces of few colours

In 1995, Erd\H{o}s and Gy'arf'as proved that in every $2$-colouring of the edges of $K_n$, there is a vertex cover by $2\sqrt{n}$ monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers $r,s$, what is the smallest number ${\text{pc}}_{r,s}(K_n)$ such that in every colouring of the edges of $K_n$ with $r$ colours, there exists a vertex cover of $K_n$ by ${\text{pc}}_{r,s}(K_n)$ monochromatic paths using altogether at most $s$ different colours? For fixed integers $r>s$ and as $n\to \mathrm{\infty }$, we prove that $$\text{pc}$_{r,s}(K_n) = \Theta(n^{1/\chi})$,where$\chi=\max{{1,2+2s-r}}$ is the chromatic number of the Kneser gr aph $\text{KG}(r,r-s)$. More generally, if one replaces $K_n$ by an arbitrary $n$-vertex graph with fixed independence number $\alpha$, then we have ${\text{pc}}_{r,s}(G)=O(n^{1\mathrm{/}\chi })$, where this time around $\chi$ is the chromatic number of the Kneser hypergraph ${\text{KG}}^{(\alpha +1)}(r,r-s)$. This result is tight in the sense that there exist graphs with independence number $\alpha$ for which ${\text{pc}}_{r,s}(G)=\mathrm{\Omega }(n^{1\mathrm{/}\chi })$. This is in sharp contrast to the case $r=s$, where it follows from a result of S'ark"ozy (2012) that ${\text{pc}}_{r,r}(G)$ depends only on $r$ and $\alpha$, but not on the number of vertices. We obtain similar results for the situation where instead of using paths, one wants to cover a graph with bounded independence number by monochromatic cycles, or a complete graph by monochromatic $d$-regular graphs.