Let $X = \{1,\ldots,2 \cdot n\}$ be a set of numbers. For this question, an *equal* partition $Y_1,Y_2$ of $X$ is a partition of $X$ to two equal sizes. Let $\varepsilon \in (0,1)$ be an error parameter. Define an $\varepsilon$-*cover* of $X$ of size $k$ as equal partitions $(Y^i_1,Y^i_2)_{i \in \{1,\ldots,k\}}$ such that the following holds: For any equal partition $Z_1, Z_2$ of $X$, there is $i \in \{1,\ldots,k\}$ such that $|Z_1 \cap Y^i_1| \geq (1-\varepsilon) \cdot n$. That is, an $\varepsilon$-cover contains a good approximation for any equal partition.

My question is what is the smallest size of an $\varepsilon$-cover? Can it be of size independent of $n$? can it be polynomial in $1/\varepsilon$?

This seems like a fundamental combinatorial question and I assume variants of this problem have been studied. However, I am unfamiliar with the right keywords to find such results. Any form of help would be very appreciated.