Monochromatic connected matchings in 2-edge-colored multipartite graphs

An improvement on Łuczak's connected matchings method

“…We note that while Lemma 1 is stated for the particular case of the k-colour Ramsey number of paths and even cycles, a similar statement can be obtained for various problems of similar nature. For example, the original lemma in [9] can be used to study asymmetric Ramsey numbers of paths and even cycles, namely where we require different path or cycle lengths for different colours 1 . Moreover, it is applicable to odd cycles, where an ℓ-coloured C n , for odd n, corresponds to an ℓcoloured connected matching of size n/2 that is contained in a non-bipartite ℓ-coloured component.…”