For given graphs G?,G?,...,Gt, the multi color Ramsey number R(G?,G?,...,Gt) is the smallest positive integer n such that if the edges of the complete graph Kn are partitioned into t disjoint color ?n??, i.e.,when n is odd and large enough,every ?-coloring of the edges of the complete graph on ?n?? vertices, K_(?n??), contains a monochromatic Cn. Hear we prove a Ramsey-type problem with the similar result. Consider a graph G on N = ?n?? vertices,where n is sufficiently large odd integer and let ?(G) gt; ?N/?. We show asymptotically that every ?-coloring of the edges of G, contains a monochromatic Cn. More pricisely, we show that for every ? gt; ? there exists n? such that for every odd n ? n? and every graph G on (? amp;#??;?)n vertices with ?(G) ? (?/? amp;#??;?)n, each colouring of the edges of G with three colours leads to a monochromatic cycle of length n. For the proof we show that in every ?-coloring of the edges of G there exists a color which contains a matching saturating at least n vertices contained in a component in this color which is non-bipartite. Then using Szemerédi’s regularity lemma we will show how existence of Cn follows from existence of this m
