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 $t \to \infty$, \[g(t) = \Theta(t^{ 1+o(1)}).\]
Categories: math.CO