Frankl’s union-closed sets conjecture: raise the proven constant from ≈0.3829 to 1/2
Simple to state, open for 45 years, and the entropy method now saturates near 0.383, well below 1/2, so 66: below sunflowers (71), whose best bound is also stuck one idea away. Application value 3.8 and leverage 22 are low: the entropy technique has been reused, but little depends on the answer.
The problem
In any family of sets closed under merging, must some element appear in at least half of them? Best proven: about 38 percent.
- 1A family of sets closed under merging
- 2Is some element in at least half?
- 3Entropy argument gives a fixed share
- 4Best proven is about 38 percent, not 50
Statement. Let be a finite family of finite sets, not all empty, closed under pairwise union (). says that some element lies in at least of the sets. Let be the largest constant such that some element always lies in at least sets. The problem asks to raise the best proven lower bound on toward , or to prove .
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Step 3
- SAT solving (CDCL)Algorithm
- Step 4
- Extremal and probabilistic combinatoricsTheory · host
- Then
- Frankl’s union-closed sets conjecture: raise the proven constant from ≈0.3829 to 1/2The problem
State of the art
Partially resolved: a universal constant is known, but not 1/2. Knill (1994) and Wójcik (1999) gave bounds of order 1/log₂|F|. In November 2022 Gilmer proved a constant 0.01 by an entropy argument comparing the union of two independent samples with one sample. Within days Chase–Lovett, Alweiss–Huang–Sellke, Sawin and Pebody raised it to (3−√5)/2 ≈ 0.38197; Sawin also disproved a stronger conjecture of Gilmer and showed the constant can be exceeded. Liu (arXiv:2306.08824) proved a bound above 0.3823455 with conditionally i.i.d. couplings, and his numerics suggested about 0.382709. In September–October 2026, Jiang (arXiv:2609.08291, computer-assisted) and Costa–Sadhu (arXiv:2610.02295) proved c_UC ≥ 0.38288525; Costa–Sadhu show that the inequality they use cannot exceed 0.3828852599…. These 2026 results are unrefereed. The entropy method cannot reach 1/2 as it stands.
Realistic applications
None practical. The entropy method for extremal set problems and the coupling techniques have been reused in other extremal problems.
What counts as solved
A refereed proof of c_UC = 1/2, or of any c_UC above the 0.3828852599… limit of the current entropy inequality. Small improvements within the existing framework are partial progress.
How it gets verified
Lean proofA formal proof checked by Lean 4 / Mathlib.
Statement: UnionClosed.union_closed in Wikipedia/UnionClosed.lean (Formal Conjectures pinned at commit e754a78 (17 October 2026)): every union-closed family other than {∅} has an element in at least half of its sets, with Yu's constant and other solved variants stated alongside. A raised constant c is the same statement with 1/2 replaced by c. A proof is accepted when it compiles against the pinned corpus and passes Comparator; the entropy arguments need real-analysis estimates formalized along the way.
Claims under verification
+ add a claimWhere the work is
A new method is needed; computation optimizes entropy inequalities.
Bottlenecks
ranked by votes- #1TheoryEntropy method saturates at 0.3828852...
Costa and Sadhu show the inequality used cannot exceed this value, so a different argument is required.
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