Union closed sets conjecture

Description of constant

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

Known upper bounds

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

Known lower bounds

Bound	Reference	Comments
$0$	Trivial
$0.1$	[G2022]
$(3-\sqrt{5})/2 = 0.381966\dots$	[AHS2022], [CL2022], [P2022], [S2022]
$>(3-\sqrt{5})/2$	[S2022]
$0.38234$	[C2022], [Y2022]
$0.38271$	[L2023]

Additional comments and links

A survey of the literature up to 2015 can be found at [BS2015].
This blog post by Matthew Aldridge covers more recent developments.

References

[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.