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OP-12003OpenCombinatorics

Union-closed sets conjecture constant (C6C_{6})

Union-closed sets conjecture constant: determine C6C_{6} or improve its bounds; the repository records 0.38271 (lower) and 0.5 (upper).

MSC 05D05 · Extremal set theory

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

C6C_{6} is the largest constant such that any union-closed family of sets on an NN-element ground set has an element contained in at least C6C_{6} fraction of the sets.

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Background

References (from the repository page)

  • [AHS2022] Alweiss, Ryan; Huang, Brice; Sellke, Mark. Improved lower bound for the union-closed sets conjecture. arXiv preprint arXiv:2211.11731, 2022.
  • [BS2015] Bruhn, Henning; Schaudt, Oliver. The journey of the union-closed sets conjecture. Graphs and Combinatorics, 31(6):2043-2074, 2015.
  • [C2022] Cambie, Stijn. Better bounds for the union-closed sets conjecture using the entropy approach. arXiv preprint arXiv:2212.12500, 2022.
  • [CL2022] Chase, Zachary; Lovett, Shachar. Approximate union closed conjecture. arXiv preprint arXiv:2211.11689, 2022.
  • [F1995] Frankl, Péter. Extremal set systems. Handbook of combinatorics, 2:1293-1329, 1995.
  • [G2022] Gilmer, Justin. A constant lower bound for the union-closed sets conjecture. arxiv:2211.09055, 2022.
  • [L2023] Liu, Jingbo. Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling. arXiv preprint arXiv:2306.08824, 2023.
  • [P2022] Pebody, Luke. Extension of a method of Gilmer. arXiv preprint arXiv:2211.13139, 2022.
  • [S2022] Sawin, Will. An improved lower bound for the union-closed set conjecture. arXiv preprint arXiv:2211.11504, 2022.
  • [Y2022] Yu, Lei. Dimension-Free Bounds for the Union-Closed Sets Conjecture. arXiv preprint arXiv:2212.00658, 2022.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0.38271; best known upper bound 0.5. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
1/2=0.51/2 = 0.5 [F1995] Conjectured (in 1976) to be optimal

Known lower bounds

Bound Reference Comments
00 Trivial
0.10.1 [G2022]
(3−5)/2=0.381966…(3-\sqrt{5})/2 = 0.381966\dots [AHS2022], [CL2022], [P2022], [S2022]
>(3−5)/2>(3-\sqrt{5})/2 [S2022]
0.382340.38234 [C2022], [Y2022]
0.382710.38271 [L2023]
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Union-closed sets conjecture constant ($C_{6}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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