Suppose $\mathcal{H}=(V,\mathcal{E})$ is a $3$-uniform hypergraph on $n$ vertices, such that for any non-empty vertex set $S\subseteq V$ with $|S|\le t$ (where $(t<n)$), there is a $v\in S$ and an edge $E\in\mathcal{E}$ such that $E\cap S=\{v\}$.
Now my question is: is there an $0<\alpha<1-\frac{t}{n}$, such that we can find a subset $U\subset V$ satisfying "$|U|\ge(1-\alpha)n$, and for any $U_1\subseteq U$, there exists a $v\in U_1$ an edge $E\in\mathcal{E}$ such that $E\cap U_1=\{v\}$ " ? If there is, how large can $U$ be? (I want it to be as large as possible.)