This is a naive question that could justifiably be quickly closed.
Nevertheless:


***Q***. Why is
[Péter Frankl's](https://en.wikipedia.org/wiki/Union-closed_sets_conjecture)
conjecture so difficult?



> 
> If any two sets in some family of sets have a union that also belongs to the family, must some element belong to at least half of the sets in the family?
> 
> 
> 


This has remained unsolved for ~$40$ years.
It seems that, unlike other conjectures (say, concerning prime conjectures), it has not been confirmed for vastly huge sets (just: families of at most $50$ sets).


More specifically, can anyone indicate why this conjecture seems so
difficult to prove or disprove? Why it has withstood assaults so long?



Why is the Frankl conjecture hard?
(**Migrated** by request from the comments.)


Bruhn and Schaud's (2013) [**The journey of the union-closed sets conjecture**](https://arxiv.org/abs/1309.3297) provides a rather readable write-up. Particularly relevant is the section **Obstacles to a proof**; for example, you may check just after *Conjecture 15* in which the authors ask (essentially) your question here:



> 
> "So, why then has the conjecture withstood more than twenty years of proof attempts?" (p. 14)
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Bruhn and Schaud then list three possible techniques of proof, and go into a bit of detail around why they do not seem to work out; these techniques are: *injections*, *local configurations*, and *averaging*.


The paper also provides a few relevant re-formulations using, e.g., lattices, (maximal stable sets of bipartite) graphs, and the "Salzborn" formulation (p. 12). In each case, a re-formulation of the **Frankl** (or **union-closed sets**) **conjecture** brings corresponding ideas and techniques with varying potential; the authors of this particular survey do well by their promise early on:



> 
> "The focus of this survey is on the methods employed to attack the conjecture. Our treatment of the literature is therefore somewhat uneven. Whenever we can identify a technique that, to our eyes, seems interesting and potentially powerful we discuss it in greater detail" (p. 3).
> 
> 
> 


