New bounds for a hypergraph Bipartite Tur\'an problem

Let $t$ be an integer such that $t\geq 2$. Let $K_{2,t}^{(3)}$ denote the triple system consisting of the $2t$ triples $\{a,x_i,y_i\}$, $\{b,x_i,y_i\}$ for $1 \le i \le t$, where the elements $a, b, x_1, x_2, \ldots, x_t,$ $y_1, y_2, \ldots, y_t$ are all distinct. Let $ex(n,K_{2,t}^{(3)})$ denote the maximum size of a triple system on $n$ elements that does not contain $K_{2,t}^{(3)}$. This function was studied by Mubayi and Verstra\"ete, where the special case $t=2$ was a problem of Erd\H{o}s that was studied by various authors. Mubayi and Verstra\"ete proved that $ex(n,K_{2,t}^{(3)})<t^4\binom{n}{2}$ and that for infinitely many $n$, $ex(n,K_{2,t}^{(3)})\geq \frac{2t-1}{3} \binom{n}{2}$. These bounds together with a standard argument show that $g(t):=\lim_{n\to \infty} ex(n,K_{2,t}^{(3)})/\binom{n}{2}$ exists and that $$\frac{2t-1}{3}\leq g(t)\leq t^4.$$ Addressing the question of Mubayi and Verstra\"ete on the growth rate of $g(t)$, we prove that as $t \to \infty$, $$g(t) = \Theta(t^{1+o(1)}).$$