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

In this paper we study a multi-partite version of the Erd{o}s--Stone theorem. Given integers $r<k$ and t≥ 1, let exₖ(n, Kᵣ₊₁(t)) be the maximum number of edges of Kᵣ₊₁(t)-free k-partite graphs with n vertices in each part, where Kᵣ₊₁(t) is the t-blowup of Kᵣ₊₁. An easy consequence of the supersaturaion result gives that exₖ(n, Kᵣ₊₁(t)) = exₖ(n, Kᵣ₊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 Kᵣ₊₁-free multi-partite graphs.