I have this friend who always tells me problems I can't solve. Here is the latest one. We are given a family $\mathcal F$ of at least $2^{n-1}+1 $ subsets $[n]$. We must prove that we can $\color{blue}{\text{select}}$ $\lceil \log_2 n \rceil +1$ of the subsets in $\mathcal F$ such that if we pick two elements of $[n]$ there is a $\color{blue}{\text{selected}}$ subset that contains exactly one of those two elements.
Here are some observations: The bound is sharp. If $\mathcal F$ has $2^{n-1}$ subsets we can let $\mathcal F$ be the family of subsets of $[n]$ that have both $n$ and $n-1$ or none of those two elements and then no selection satisfies the problem. I think this is related to this problem