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One natural extension of classical Ramsey numbers to multipartite graphs is to consider 2-colorings of the complete multipartite graph consisting of \( n \) parts, each of size \( k \), denoted \( K_{n \times k} \). We may then ask for the minimum integer \( n \) such that \( K_{n \times k} \rightarrow (G, H) \) for two given graphs \( G \) and \( H \). We study this number for the cases when \( G \) and \( H \) are paths or cycles and show some general bounds and relations to classical Ramsey theory.