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.

Let P n represent the path of size n. Let K 1,m−1 represent a star of size m and be denoted by S m . Given a two coloring of the edges of a complete graph K j×s we say that K j×s → (P n , S m+1 ) if there is a copy of P n in the first color or a copy of S m+1 in the second color. The size Ramsey multipartite number m j (P n , S m+1 ) is the smallest natural number s such that K j×s → (P n , S m+1 ).Given j, n, m if s = n + m − 1 − k j − 1 , in this paper, we show that the size Ramsey numbers m j (P n , S m+1 ) is bounded above by s for k = n − 1 j . Given j ≥ 3 and s, we will obtain an infinite class (n, m) that achieves this upper bound s. In the later part of the paper, will also investigate necessary and sufficient conditions needed for the upper bound to hold.

No abstract