Summary: For given two graphs \(G_1\) and \(G_2\) and integer \(j \geq 2\) the size multipartite Ramsey numbers \(m_j(G_1. G_2) = t\) is the smallest integer such that every factorization of graph \(K_{j \times t} := F_1 \oplus F_2\) satisfies the following condition: either \(F_1\) contains \(G_1\) as a subgraph or \(F_2\) contains \(G_2\) as a subgraph. In this paper, we establish exact value of the size multipartite Ramsey number \(m_j(P_n, C_s)\) for all integers \(j, n \geq 2\) and \(s = 3\) or 4, where \(P_n\) is a path on \(n\) vertices and \(C_3\) is a cycle on three vertices.