Tripartite Zarankiewicz numbers and norm graphs

For fixed integers $s\ge t\ge 2$, let $\mathrm{ex}(n,n,n,K_{s,t})$ denote the maximum number of edges in a tripartite $K_{s,t}$-free graph with $n$ vertices in each part. When $s\ge (t-1)!+1$, let $r$ be the largest integer satisfying $s\ge (t-1)!r^{t-1}+1$. Using the quotient norm graphs of Alon, R'onyai and Szab'o, we prove that $$\mathrm{ex}(n,n,n,K_{s,t})\ge (\frac{3}{2^{1\mathrm{/}t}}{r}^{1-1\mathrm{/}t}+o(1))n^{2-1\mathrm{/}t}\mathrm{.}$$ Improving an upper bound of Tait and Timmons, we prove that, for all $s\ge t\ge 2$, $$\mathrm{ex}(n,n,n,K_{s,t})\le (\frac{3}{2^{1\mathrm{/}t}}(s-t+1{)}^{1\mathrm{/}t}+o(1))n^{2-1\mathrm{/}t}\mathrm{.}$$ Together, these bounds recover the results for $t=2$, and give the new asymptotic formula $$\mathrm{ex}(n,n,n,K_{3,3})=(\frac{3}{\sqrt[3]{2}}+o(1))n^{5\mathrm{/}3}\mathrm{.}$$ Analogous results extend to $k$-partite graphs containing no $K_{s,t}$ whose $s$-vertex or $t$-vertex side lies in a single part. As an application of our tripartite construction, we determine the tripartite multicolor Ramsey number of $K_{3,3}$ asymptotically.