Frankl's Union-Closed Sets Conjecture
In any finite union-closed family of at least one nonempty set, does some element appear in at least half of the member sets?
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- Validated increment
Gilmer entropy bound: some element in ≥ 38.1% of sets
Gilmer's breakthrough replaced decades of slow constant-chasing with an entropy argument, proving a positive fraction strictly above 0. All later constant improvements sit on this foundation; the 50% conjecture remains open.
Previous best
Element frequency ≥ ε for tiny explicit ε
New certified
Element frequency ≥ 0.381… (Gilmer; refined by followers)
Justin Gilmer and subsequent refinements
Also open