Lower bounds for the union-closed sets constant
the union-closed constant
- steps
- 4
- by AI
- 0
- since
- 2022
The line is the frontier over time, climbing as the bound goes up: higher is better here. Filled dots are steps that moved it; muted dots are results that did not. Orange dots are catalog entries, results with AI in the loop. Hollow dots are candidates under review and never move the line.Grey dots along the bottom edge are results from before the quantity had a number, placed at the worst end because they have no value on this axis. Dots that would overlap are nudged sideways a few pixels. Hover a dot for its value and attribution.
About this frontier
Frankl conjectured in 1979 that every finite union-closed family of sets, not all empty, has an element belonging to at least half its members. Until 2022 no constant at all was known: nobody could name a with an element in at least a fraction. Gilmer's entropy argument broke that, and the constant has climbed since. This frontier tracks the best proved lower bound. Its finish line is exactly , which would settle Frankl. It also has a ceiling: the entropy method in every form used since 2022 provably cannot certify more than , so reaching needs a different idea rather than a better constant.
Every step, newest first
| Date | Value | Who | Model | Status | Source |
|---|---|---|---|---|---|
| 8 Sept 2026 | Computer-assisted and conditional on two numerically verified hypotheses of the same kind as those behind Liu's record, so it is a candidate rather than the headline. The same work proves the ceiling that ends this staircase: no certificate of this form can exceed 0.382885260. | Andrew Moffat with Claude Fable 5.1 | Fable 5.1 | candidate lean-checked | entry |
| 15 Jun 2023 | best Conditionally i.i.d. coupling, building on Yu, Sawin and Gilmer. The standing record, and the last movement for three years. | Jingbo Liu | – | AI step | source ↗ |
| 23 Dec 2022 | Past the barrier that (3-sqrt 5)/2 appeared to be, by using the dependent samples Sawin suggested. Solves Sawin's question exactly, and the same paper studies where the approach runs out. | Stijn Cambie | – | historical | source ↗ |
| 21 Nov 2022 | The limit Gilmer conjectured his own method would reach, proved five weeks later. Chase and Lovett, Sawin, and Pebody obtained the same constant independently within days; this row credits one of four simultaneous proofs and the others are not disputed. | Alweiss, Huang and Sellke | – | historical | source ↗ |
| 16 Nov 2022 | The first constant bound of any size, by an entropy argument. Before this paper no positive constant was known at all, which is why a bound of 0.01 was the breakthrough rather than the number it names. | Justin Gilmer | – | historical | source ↗ |
Steps read at source on arXiv. The (3-sqrt 5)/2 row is credited to Alweiss, Huang and Sellke, but Chase and Lovett, Sawin, and Pebody proved it independently within days; the row's note says so. The 2026 row is a candidate: computer-assisted and conditional on two numerically verified hypotheses, so it is drawn hollow and is not the headline, which stays Liu's 0.382709. The ceiling at 0.382885260 is a theorem about the method rather than a value reached, so it is stated here and in the statement rather than drawn as a step.