Turán Number of Disjoint Triangles in 4-Partite Graphs

Let k ≥ 2 and n₁ ≥ n₂ ≥ n₃ ≥ n₄ be integers such that n₄ is sufficiently larger than k. We determine the maximum number of edges of a 4-partite graph with parts of sizes n₁, …, n₄ that does not contain k vertex-disjoint triangles. For any r > t ≥ 3, we give a conjecture on the maximum number of edges of an r-partite graph that does not contain k vertex-disjoint cliques K_t.