Tur\'an number of complete multipartite graphs in multipartite graphs

In this paper we study a multi-partite version of the Erdos--Stone theorem. Given integers r<k and t 1, let exₖ (n, Kₑ+₁ (t) ) be the maximum number of edges of Kₑ+₁ (t) -free k-partite graphs with n vertices in each part, where Kₑ+₁ (t) is the t-blowup of Kₑ+₁. An easy consequence of the supersaturaion result gives that exₖ (n, Kₑ+₁ (t) ) = exₖ (n, Kₑ+₁) +o (n²). Similar to a result of Erd os and Simonovits for the non-partite case, we find that the error term is closely related to the (multi-partite) Zarankiewicz problem. Using such Zarankiewicz numbers, for t=2, 3, we determine the error term up to an additive linear term; using some natural assumptions on such Zarankiewicz numbers, we determine the error term up to an additive constant depending on k, r and t. We actually obtain exact results in many cases, for example, when k 0, 1 r. Our proof uses the stability method and starts by proving a stability result for Kₑ+₁-free multi-partite graphs.