Tiling Edge-Coloured Graphs with Few Monochromatic Bounded-Degree Graphs

We prove that for all integers \(\Delta ,r \ge 2\), there is a constant \(C = C(\Delta ,r) >0\) such that the following is true for every sequence \({\mathcal {F}}= \{F_1, F_2, \ldots \}\) of graphs with \(v(F_n) = n\) and \(\Delta (F_n) \le \Delta \), 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.