I have posted this problem elsewhere, but I don't think anyone has seriously tried it yet.

Resolves YES if there is a solution posted in the comments to this market that I can understand and I am satisfied with being correct by market close time (one week from market creation time).

For reference, it took me about 3-4 hours of partially-focused time to solve it.

Here's the problem:

Let $S=\{1,2,\dots,n\}$ and let $B$ be a family of subsets of $S$. Let $A$ be the family of subsets of $S$ that are a (not necessarily proper) superset of some set in $B$. Suppose that $|B|\le 100n$. You are told $n$ but not $A$ or $B$.

In a \textit{query}, you may ask about one subset of $S$, and you will be told whether it belongs in $A$.

Does there exist a strategy that takes a subexponential number of queries and finds a set in $B$ with minimum size?

Since there is some amount of subjective judgment on my side for this market, I will not bet.