Frankl's union-closed sets conjecture

If a family of sets contains the union of any two of its members, must some element belong to at least half of the sets? Frankl's conjecture says yes. Since 2022 an entropy method has proved the fraction is at least about 0.38, and closing the gap to 1/2 is one of the best-known open problems in extremal combinatorics.

Problem

A family $\mathcal{F}$ of subsets of a finite set is union-closed if $A \cup B \in \mathcal{F}$ whenever $A, B \in \mathcal{F}$. Let $C$ be the largest constant such that every union-closed family $\mathcal{F} \ne \{\emptyset\}$ has an element contained in at least $C\,|\mathcal{F}|$ of its sets.

The power set of $\{1, \dots, n\}$ is union-closed and each element lies in exactly half of its sets, so $C \le 1/2$. The union-closed sets conjecture asserts $C = 1/2$. Determine $C$, or improve the lower bound.

Background

The conjecture dates from 1979 and is attributed to Péter Frankl. For four decades the best general bounds decayed with the size of the family. In 2022 Gilmer proved the first constant lower bound with an information-theoretic argument: if every element is rare in a random member $A$ of the family, then the union of two independent random members has strictly more entropy than $A$. Within days several groups sharpened his inequality to $(3-\sqrt{5})/2$, and later work has refined the couplings used in the argument.

What is known

• Lower bounds: $0.01$ (Gilmer, 2022, the first constant bound); $(3-\sqrt{5})/2 \approx 0.38197$ (Alweiss, Huang and Sellke; Chase and Lovett; Pebody; Sawin; all 2022); strictly larger than $(3-\sqrt{5})/2$ (Sawin, 2022); about $0.38234$ (Cambie, 2022; Yu, 2022); a strict improvement on $0.38234$, proved analytically, and about $0.38271$ under hypotheses that are verified numerically rather than proved (Liu, 2023).
• Upper bound: $1/2$, from the power set; this is the conjectured value.
• Small cases: the conjecture holds when the ground set has at most 12 elements (Vučković and Živković, 2017). Roberts and Simpson (2010) showed that a counterexample needs at least $4q - 1$ sets, where $q$ is the smallest ground-set size of a counterexample, so the conjecture also holds for families of at most 50 sets.

What would count as progress

• An explicit rigorous bound above $0.38234$, for example by proving Liu's numerical hypotheses with interval arithmetic so that $0.38271$ becomes a theorem, or a new coupling that beats $0.38271$. Such proofs reduce to finite-dimensional optimization problems whose certificates can be rerun.
• A proof of the conjecture, $C = 1/2$. A Lean statement is in formal-conjectures.
• A counterexample: an explicit union-closed family in which every element lies in fewer than half of the sets. Checking it takes milliseconds (verify closure under unions and count each element).

References

1. J. Gilmer, "A constant lower bound for the union-closed sets conjecture" (2022). arXiv:2211.09055
2. R. Alweiss, B. Huang and M. Sellke, "Improved lower bound for Frankl's union-closed sets conjecture" (2022). arXiv:2211.11731
3. W. Sawin, "An improved lower bound for the union-closed set conjecture" (2022). arXiv:2211.11504
4. J. Liu, "Improving the lower bound for the union-closed sets conjecture via conditionally IID coupling" (2023). arXiv:2306.08824
5. I. Roberts and J. Simpson, "A note on the union-closed sets conjecture", Australasian Journal of Combinatorics 47 (2010), 265–267. PDF
6. H. Bruhn and O. Schaudt, "The journey of the union-closed sets conjecture", Graphs and Combinatorics 31 (2015), 2043–2074. doi:10.1007/s00373-014-1515-0

Source

Imported from Optimization Constants in Mathematics, constant 6, curated by Terence Tao and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.