I will discuss Ramsey and Ramsey–Turán type problems for matchings. In particular, I will answer the following question: among all graphs that admit an edge colouring with no monochromatic matching of a given size, which ones have the maximum number of triangles (or, more generally, copies of a fixed clique)? This question generalises both the Ramsey number of matchings (Cockayne–Lorimer) and the Turán number of matchings (Erdős–Gallai), together with its clique-count variant. Our proof identifies two explicit extremal constructions and uses a compression-based optimisation algorithm that relies on the Gallai–Edmonds decompositions in each colour. Time permitting, I will discuss additional applications of this algorithm.

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