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

Abstract

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 $$\underset{n\to \infty }{\lim }\frac{ex(n,n,n,K_{2,t})}{n^{3/2}}=\frac{3\sqrt{t-1}}{\sqrt{2}},\text{for integer }t\ge 2.$$