On the Precise Asymptotics of $ex(n,n,n,K_{2,t})$ for even $t$

Let $K_{2,t}$ denote the complete bipartite graph. For an integer $n\ge 1$, let $ex(n,n,n,K_{2,t})$ be the maximum number of edges in an $n\times n\times n$ tripartite graph (that is, a 3-partite graph with three parts each of size $n$) containing no copy of $K_{2,t}$. In this paper we prove that, for even $t\ge 2$, $$ex(n,n,n,K_{2,t}) \ge \frac{3\sqrt{t-1}}{\sqrt{2}}\, n^{3/2} + o(n^{3/2}).$$ Combining our construction with earlier work of Tait and Timmons, we obtain $$\lim\limits_{n\to\infty} \frac{ex(n,n,n,K_{2,t})}{n^{3/2}} = \frac{3\sqrt{t-1}}{\sqrt{2}}, \qquad\text{for integer } t\ge 2.$$