On characterizing the critical graphs for matching Ramsey numbers

Given simple graphs H₁,H₂,…,H_c, the Ramsey number r(H₁,H₂,…,H_c) is the smallest positive integer n such that every edge-colored K_n with c colors contains a subgraph in color i isomorphic to H_i for some i∈{1,2,…,c}. The critical graphs for r(H₁,H₂,…,H_c) are edge-colored complete graphs on r(H₁,H₂,…,H_c)−1 vertices with c colors which contain no subgraphs in color i isomorphic to H_i for any i∈{1,2,…,c}. For n₁≥n₂≥⋯≥n_c≥1, Cockayne and Lorimer (1975) showed that r(n₁K₂,n₂K₂,…,n_cK₂)=n₁+1+∑i=1c(n_i−1), in which n_iK₂ is a matching of size n_i. Using the Gallai–Edmonds Theorem, we characterized all the critical graphs for r(n₁K₂,n₂K₂,…,n_cK₂), implying a new proof for this Ramsey number.