An exact robust Ramsey theorem for matchings

Abstract

Keevash and Michaeli recently proved that, under the robustness assumption that \(G\) is an \(s\)-connector (i.e. \(\overline G\) 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 \({\bf t}=(t_1,\ldots,t_q)\in\mathbb N_+^q\), let \(R_s({\bf t})\) be the smallest integer \(N\) such that every \(N\)-vertex \(s\)-connector \(G\) satisfies \( G\to (t_1K_2,\ldots,t_qK_2). \) We determine the exact value \[ R_s({\bf t})=\sum_{j\in[q]}(t_j-1)+ \max\left\{2s,\ s+\max_{j\in[q]}t_j\right\}. \] 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.