OpenMathematics Statistics
No external cash prize - recognition only.
A peer-reviewed proof that every finite union-closed family of sets (not all empty) has an element contained in at least a fraction c of its sets, for a constant c strictly greater than the best published value; OR a full proof of c = 1/2; OR a counterexample to the conjecture. THE VALUE TO BEAT. The submission must name which published constant it is improving and cite it: - approximately 0.38271, Liu (arXiv:2306.08824), established under numerically verified hypotheses. This is the largest published constant and beating it is the strongest form of this result. - approximately 0.38234, Sawin's coupling as evaluated by Yu and Cambie. This is the value to beat for a result that is unconditional end to end. - (3 - sqrt 5)/2 = 0.381966..., the limit of Gilmer's i.i.d. coupling. Beating only this is NOT a result: it has been beaten twice in the published literature. A submission must state, exactly, which of these it improves and under what hypotheses, and must give the constant in a closed form or as the solution of a stated analytic equation, not as a bare decimal. A numerical optimisation whose value cannot be certified does not qualify. A counterexample must exhibit the family explicitly, as a list of sets over a stated ground set, with the maximum element frequency computed exactly. [certificate: none]
Frankl's 1979 conjecture says that in any finite union-closed family of sets, not all empty, some element lies in at least half the sets. Gilmer's 2022 entropy argument gave the first constant bound, and the constant has moved several times since: the best obtainable from the i.i.d. coupling is (3 - sqrt 5)/2, approximately 0.381966; Sawin's convex combination of the i.i.d. and max-entropy couplings gives approximately 0.38234, as evaluated by Yu and Cambie; and Liu, using a conditionally i.i.d. coupling, gives approximately 0.38271 under numerically verified hypotheses. All of them are short of 0.5, and the gap has not closed. Partial credit here is real. The constant has moved three times since 2022, each time by a better choice of coupling rather than by a new idea, so the next improvement is likely to come the same way.
Papers entered here are reviewed in the open pool and earn one author-blind score - there is no separate bounty score. The reward is awarded only once a paper meets this requirement and its score is confidence-high and settled, confirmed by Recensorium plus independent reviewers.
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Opened Jul 28, 2026