An exact robust Ramsey theorem for matchings

Keevash and Michaeli recently proved that, under the robustness assumption that $G$ is an $s$-connector (i.e. $\bar{G}$ is $K_{s,s}$-free), $G$ has essentially the same multicolour Ramsey matching properties as complete graphs, with an additive error $O(qs)$, where $q$ is the number of colours. They asked whether the dependence on $q$ can be removed. We answer this question in a stronger exact form. For $\mathbf{t}=(t_1,\dots ,t_q)\in {\mathbb{N}}_{+}^q$, let $R_s(\mathbf{t})$ be the smallest integer $N$ such that every $N$-vertex $s$-connector $G$ satisfies $G\to (t_1{K}_2,\dots ,t_q{K}_2)\mathrm{.}$ We determine the exact value $$R_s(\mathbf{t})=\sum _{j\in [q]}(t_j-1)+\max \{2s,s+\underset{j\in [q]}{\max }{t}_j\}\mathrm{.}$$ While Keevash and Michaeli’s proof uses a compression algorithm based on the Gallai–Edmonds decomposition to reduce the colouring to a structured form, our proof is a direct minimal-counterexample argument together with a new counting method for monochromatic matchings which can be applied to $s$-connectors.