The Size Multipartite Ramsey Numbers for Paths

For graphs \( G_1, G_2, \ldots, G_k \), the (generalized) \text{size multipartite Ramsey number} \( m_j(G_1, G_2, \ldots, G_k) \) is the least natural number \( m \) such that any coloring of the edges of \( K_{j \times m} \) with \( k \) colors will yield a copy of \( G_i \) in the \( i \)th color for some \( i \). In this note, we determine the exact value of the size multipartite Ramsey number \( m_j(P_s, P_t) \) for \( s = 2, 3 \) and all integers \( t \geq 2 \), where \( P_t \) denotes a path on \( t \) vertices.