Tiling edge-coloured graphs with few monochromatic bounded-degree graphs

Abstract

We prove that for all integers $\hat{I}”,r\ge 2$, there is a constant $C=C(\hat{I}”,r)>0$ such that the following is true for every sequence $\mathcal{F}=\{F_1,F_2,\dots \}$ of graphs with $v(F_n)=n$ and $\hat{I}”(F_n)\le \hat{I}”$, for each $n\in \mathbb{N}$. In every $r$-edge-coloured $K_n$, there is a collection of at most $C$ monochromatic copies from $\mathcal{F}$ whose vertex-sets partition $V(K_n)$. This makes progress on a conjecture of Grinshpun and Sárközy.

26 pages, 1 figure