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arXiv · 2109.09642

Tiling with monochromatic bipartite graphs of bounded maximum degree

António Girão·Oliver Janzer
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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.

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António Girão, Oliver Janzer. 2021-09-20. Tiling with monochromatic bipartite graphs of bounded maximum degree. https://arxiv.org/abs/2109.09642

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