Let the star on n vertices, namely K1,n−1 be denoted by Sn. If every two coloring of the edges of a complete balanced multipartite graph Kj×s there is a copy of Sn in the first color or a copy of Sm in the second color, then we will say Kj×s → (Sn, Sm). The size Ramsey multipartite number mj(Sn, Sm) is the smallest natural number s such that Kj×s → (Sn, Sm). In this paper, we obtain the exact values of the size Ramsey numbers mj(Sn, Sm) for n, m 3 and j 3.

For simple graphs G 1 and G 2 , the size Ramsey multipartite number m j (G 1 , G 2 ) is defined as the smallest natural number s such that any arbitrary two coloring of the graph K j×s using the colors red and blue, contains a red G 1 or a blue G 2 as subgraphs. In this paper, we obtain the exact values of the size Ramsey numbers m j (nK 2 , C m ) for j ≥ 2 and m ∈ {3, 4, 5, 6}.

Let G and H be two simple subgraphs of s j K ×. The smallest positive integer s such that any red and blue colouring of s j K × has a copy of red G or a blue H is called the multipartite Ramsey number of G and H. It is denoted by) , (H G m j. This paper presents exact values for) , (2 G B m j where G is a isolate vertex free graph up to four vertices.

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It has been conjectured that r(Cm, Kn) = (m − 1)(n − 1) + 1 for all m ≥ n ≥ 4. This has been proved recently for n = 4 and n = 5. In this paper, we prove that r(C5, K6) = 21. This raises the possibility that r(Cm, K6) = 5m − 4 for all m ≥ 5. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 99–108, 2000