Given a universe $U = \{e_1 , . . . , e_n\}$ of
elements, and given a collection $S = \{S_1 , . . . , S_m \}$ of subsets of $U$, each of size $\le k$, the subcollection $S' \subseteq S$ is a *unique coverage* of $V \subseteq U$ if each $e \in V$ is  uniquely covered, i.e., appears in exactly one set of $S'$. For simplicity, we assume that $\cup {S_i}=U$.

**Question 1:** Give a lowerbound on the maximum size of $V$, as a function $f(n, k)$. 

**Question 2:** Does it help if $S$ is a [Sperner family][1]?

Notice that here we are not interested in computational and algorithmic aspects. I think I can come up with a $n/k^4$ lowerbound (and constructive).

**Background**

The  problem of  maximizing the unique cover for $k \ge 3$ is NP-hard. The approximation algorithms are studied in [1]. Approximation algorithms for  Generalizations are studied in [2]. 

**References**

[1] V. Guruswami and L. Trevisan, The complexity of making unique choices: Approximating 1-in-k SAT, 2005.

[2] ERIK D. DEMAINE , URIEL FEIGE , et al, Combination can be hard: Approximability of the unique coverage problem


  [1]: https://en.wikipedia.org/wiki/Sperner_family