Given two graphs $H$ and $G$‎, ‎the size multipartite Ramsey number $ m_j (H,G )$ is the smallest natural number $t$ such that an arbitrary coloring of the edges‎ ‎of $K_{j \times t}$‎, ‎complete multipartite graph whose vertex set is partitioned into $j$ parts each of size $t$‎, ‎using two colors red and blue‎, ‎necessarily forces a red copy of $H$ or a blue copy of $G$ as a subgraph‎. ‎The notion of size multipartite Ramsey number has been introduced by Burger and Vuuren in 2004‎. ‎It is worth noting that‎, ‎this concept is derived by using the idea of the original classical Ramsey number‎, ‎multipartite‎ ‎Ramsey number and the size Ramsey number‎. ‎\\‎ ‎Recently‎, ‎we determined the order of magnitude of size multipartite Ramsey numbers of graphs $K_m$ and $K_{1,n}$ ($m_j(K_m‎, ‎K_{1,n})$)‎. ‎Indeed‎, ‎we found a lower and an upper bound for $m_j(K_m‎, ‎K_{1,n})$‎, ‎for every positive integers $j‎, ‎m$ and $n$‎. ‎We showed that the upper bound is tight‎. ‎We also found the exact values of $m_j({K_m‎, ‎K_{1,n}})$ when $m=3$‎, ‎$j=m$ and $ m-1 |\ j$‎.

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