On size multipartite Ramsey numbers of {}_{m}K_{1:n}versus P_3 or K_{1:3}

For given two simple graphs G and H; the size multipartite Ramsey num- ber mj(G;H); is the smallest natural number t such that every 2 edge coloring on the edges of the complete balanced multipartite graph Kj t has a monochromatic copy of G in the rst color or H in the second color. Hattingh and Henning (1998) gave the results for the size bipartite Ramsey numbers of stars versus paths, m2(K1;m; Pn); for m; n 2. In 2016, we have derived the size tripartite Ramsey numbers m3(mK1;n; P3); for m 1; n 2; where mK1;n is a disjoint union of m copies of a star K1;n and P3 is a path of order 3. In this paper, we determine the size multipartite Ramsey num- bers mj(mK1;n; P3) and mj(mK1;n;K1;3); for all integers m; n 2 and j 3. We also provide an exact value of m2(mK1;n; P3); for m; n 2 and m3(mK1;n;K1;3); for (m = 1; n 1) or (n = 1;m 1).