All the references I can find to Covering Set appear to be algorithmic. Is there are any reference for the simple existential question ---

Suppose we have $k$ sets $X_1,…,X_k$ which are subsets of a ground set $X$ of size $n$. We know that $|X_i| \geq t$ for all $X_i$. We would like to find the smallest set $Y$ such that $Y \cap X_i \neq \emptyset$ for all $i$.

Given fixed values for $k, n, t$ what is the largest $Y$ which might be necessary to do this?

The greedy algorithm for constructing $Y$ is to add the element of $X$ to maximize the number of $X_i$ covered in each stage. Apparently this requires $|Y| = O(\frac{n}{t} \log k)$.

If I am not concerned with an algorithm for finding $Y$, only guaranteeing that it exists, is there any asymptotically better result possible?