A new proof on the Ramsey number of matchings

For given simple graphs $H_{1},H_{2},\ldots,H_{c}$, the Ramsey number $r(H_{1},H_{2},\ldots,H_{c})$ is the smallest positive integer $n$ such that every edge-colored $K_{n}$ with $c$ colors contains a subgraph isomorphic to $H_{i}$ in color $i$ for some $i\in\{1,2,\ldots,c\}$. A critical graph of $r(H_1,H_2,\ldots,H_c)$ is an edge-colored complete graph on $r(H_1,H_2,\ldots,H_c)-1$ vertices with $c$ colors which contains no subgraph isomorphic to $H_{i}$ in color $i$ for any $i\in \{1,2,\ldots,c\}$. For $n_1\geq n_2\geq \ldots\geq n_c\geq 1$, Cockayne and Lorimer (The Ramsey number for stripes, J. Austral. Math. Soc. 19 (1975) 252--256.) showed that $r(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})=n_{1}+1+ \sum\limits_{i=1}^c(n_{i}-1)$, in which $n_{i}K_{2}$ is a matching of size $n_{i}$. In this paper, using Gallai-Edmonds Structure Theorem, we give a new proof on the value of $r(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})$ which also characterized all critical graphs