Frankl's well-known union-closed conjecture states that if F is a finite family of sets that is closed under taking unions (that is, if A and B belong to the family then so does $A\cup B$), then there must be an element that belongs to at least half the sets.

I know that pretty well any naive approach one takes to this conjecture is known to fail. By "naive approach" I suppose I mean something like an observation that it would follow from such-and-such a stronger conjecture -- it seems that all sensible stronger conjectures one thinks of are false. A very simple example of a stronger conjecture would be that if you pick a random element then on average it will belong to at least half the sets. That is completely false: take the family that consists of the empty set, {1}, and {1,2,3,4,5,6,7,8,9,10}, for example. One can try to "correct" this strengthening by devices such as insisting that for any two elements there is a set that contains one and not the other (which WLOG is the case), but such corrections don't get one very far.

What I am asking for is examples, either small ones or ones that are constructed theoretically, of union-closed families that defeat more sophisticated strengthenings of the original conjecture. I'm fairly sure they are out there but I am not an expert on this problem so I don't know them myself.

Apologies in advance if this resembles an existing question (which it feels as thought it easily might). But I've looked and not found anything.

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