Tiling with monochromatic bipartite graphs of bounded maximum degree

Abstract

We prove that for any $r\in \mathbb{N}$, there exists a constant $C_r$ such that the following is true. Let $\mathcal{F}=\{F_1,F_2,\dots \}$ be an infinite sequence of bipartite graphs such that $|V(F_i)|=i$ and $\Delta (F_i)\le \Delta$ hold for all $i$. Then in any $r$-edge coloured complete graph $K_n$, there is a collection of at most $\exp (C_r\Delta )$ monochromatic subgraphs, each of which is isomorphic to an element of $\mathcal{F}$, whose vertex sets partition $V(K_n)$. This proves a conjecture of Corsten and Mendonça in a strong form and generalizes results on the multicolour Ramsey numbers of bounded-degree bipartite graphs.

18 pages