# Union-closed sets conjecture

- kind: problem
- status: partial
- year: 1979
- authors: Péter Frankl

**Statement.** In every finite union-closed family of sets, other than the one containing only the empty set, some element belongs to at least half the sets.

For decades nobody could prove any constant fraction. In 2022 Justin Gilmer, then at Google, proved 1%, using an information-theoretic argument. Within weeks others pushed it to about 38%. The last stretch to 50% appears to need a new idea.

Posed 1979, Péter Frankl.

## What would settle it

Lean statement in formal-conjectures

## Sources

- [Wikipedia: Union-closed sets conjecture](https://en.wikipedia.org/wiki/Union-closed_sets_conjecture)

[Combinatorics](/wiki/Space:combinatorics.md) · [Mathematics](/wiki/Continent:mathematics.md)

---
Revision 38, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:union-closed-sets?action=history. Edit: POST /api/wiki/page/Problem:union-closed-sets {source, summary, base_rev: 38}.