Forbidding $K_{2,t}$ traces in triple systems

Abstract

Let $H$ and $F$ be hypergraphs. We say $H$ contains $F$ as a trace if there exists some set $S\subseteq V(H)$ such that $H{|}_S:=\{E\cap S:E\in E(H)\}$ contains a subhypergraph isomorphic to $F$. In this paper we give an upper bound on the number of edges in a $3$-uniform hypergraph that does not contain $K_{2,t}$ as a trace when $t$ is large. In particular, we show that ${\lim }_{t\to \infty }{\lim }_{n\to \infty }\frac{\mathrm{ex}(n,{\mathrm{Tr}}_3(K_{2,t}))}{t^{3/2}{n}^{3/2}}=\frac{1}{6}.$ Moreover, we show $\frac{1}{2}{n}^{3/2}+o(n^{3/2})\le \mathrm{ex}(n,{\mathrm{Tr}}_3(C_4))\le \frac{5}{6}{n}^{3/2}+o(n^{3/2})$.