Long monochromatic paths and cycles in 2-edge-colored multipartite graphs

We solve four similar problems: for every fixed s and large n, we describe all values of n₁, …, n_s such that for every 2-edge-coloring of the complete s-partite graph K_n1,…,n_s there exists a monochromatic (i) cycle C_2n with 2n vertices, (ii) cycle C_≥2n with at least 2n vertices, (iii) path P_2n with 2n vertices, and (iv) path P_2n+1 with 2n + 1 vertices. This implies a generalization for large n of the conjecture by Gyárfás, Ruszinkó, Sárközy and Szemerédi that for every 2-edge-coloring of the complete 3-partite graph K_n,n,n there is a monochromatic path P_2n+1 . An important tool is our recent stability theorem on monochromatic connected matchings.