Just read about this conjecture: Union-closed sets conjecture.
The conjecture states that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in the family. (Wikipedia)
It is quite interesting in the sense that the statement is extremely elementary (just basic set notation knowledge is enough to understand it). But it seems that even the experts can’t prove it.
One basic example is: {{1},{2},{1,2}}. The element 2 belongs to 2/3>1/2 of the sets in the family.
