Tiling with monochromatic bipartite graphs of bounded maximum degree

TL;DR

This paper proves that in any r-colored complete graph, a bounded number of monochromatic bipartite graphs with bounded maximum degree can partition the vertices, confirming a conjecture and extending multicolor Ramsey number results.

Contribution

It establishes a bound on the number of monochromatic bipartite graphs needed to partition the vertices in any r-colored complete graph, confirming a conjecture and generalizing previous results.

Findings

Existence of a constant C_r for partitioning

Bound on the number of monochromatic subgraphs

Generalization of multicolor Ramsey numbers

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 $\Delta(F_i)\leq \Delta$ 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 Mendon\c{c}a in a strong form and generalizes results on the multicolour Ramsey numbers of bounded-degree bipartite graphs.