# P41: Union-closed sets conjecture

- ID: `P41`
- Reference: `union-closed-sets-conjecture`
- Page: https://theoremdb.org/statements/P41
- Record maturity: Reviewed problem with recorded work

## Problem

If \(\mathcal{F}\) is a finite nonempty family of finite sets satisfying \(A\cup B\in\mathcal{F}\) for all \(A,B\in\mathcal{F}\), then some element belongs to at least \(|\mathcal{F}|/2\) members of \(\mathcal{F}\).

### Context

The conjecture, also called Frankl's conjecture, asks whether closure under union forces one element to appear with frequency at least one half.

### Problem setup

- **Definition (A family).** A family is union-closed when the union of any two member sets is also a member.
- **Definition (The frequency of an element).** The frequency of an element is the number of member sets containing it.
- **Remark.** The conjecture, also called Frankl's conjecture, asks whether closure under union forces one element to appear with frequency at least one half.

### What counts as a solution

- Prove the half-frequency conclusion for every qualifying finite family, or give a finite union-closed family and prove that each element occurs in fewer than half of its sets.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: Gilmer proved the first constant 0.01. Subsequent entropy work gives an unconditional constant about 0.38234. A reported 0.38271 relies on numerically verified hypotheses. Exact unresolved remainder: Prove half-frequency for every qualifying finite union-closed family, or certify a counterexample. [1](#reference-1) [2](#reference-2) [3](#reference-3)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: Gilmer proved the first constant 0.01. Subsequent entropy work gives an unconditional constant about 0.38234. A reported 0.38271 relies on numerically verified hypotheses. Exact unresolved remainder: Prove half-frequency for every qualifying finite union-closed family, or certify a counterexample.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: Gilmer proved the first constant 0.01. Subsequent entropy work gives an unconditional constant about 0.38234. A reported 0.38271 relies on numerically verified hypotheses.

Exact unresolved remainder: Prove half-frequency for every qualifying finite union-closed family, or certify a counterexample.

### Background and intake notes

- Original intake status: The cited 2024 paper identifies the union-closed sets conjecture as open. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited paper and current literature on 2026-07-22.
- Strong asymptotic lower bounds and many finite or structural cases are known, while the one-half threshold remains open.

- Recorded example: For the family {empty set, {1}, {1,2}}, the element 1 appears in two of the three sets.

### Open directions

- **Route 1** (reported): Prove the half-frequency conclusion for every qualifying finite family, or give a finite union-closed family and prove that each element occurs in fewer than half of its sets. [1](#reference-1)

### Computational notes

- Exhaustive family generation can verify bounded universe or family sizes only.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `union-closed-sets-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Kengbo Lu and Abigail Raz, “Note on the union-closed sets conjecture and Reimer's average set size theorem”. arXiv:2405.10639 (2024). arXiv:2405.10639, abstract and introduction https://arxiv.org/abs/2405.10639
   - Also cited at Abstract and introduction
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2405.10639, checked 2026-07-31; checked 2026-07-31
   - Source use: original_summary
   - The cited 2024 paper identifies the union-closed sets conjecture as open. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - Packet-linked open-status source.
   - Source named by the research packet.
2. <a id="reference-2"></a>Justin Gilmer, “A constant lower bound for the union-closed sets conjecture”. arXiv:2211.09055 (2022). Abstract https://arxiv.org/abs/2211.09055
   - preprint; primary source; arXiv:2211.09055v2; checked 2026-08-01
   - Source use: original_summary
   - First universal constant lower bound.
3. <a id="reference-3"></a>Lei Yu, “Dimension-Free Bounds for the Union-Closed Sets Conjecture”. Entropy 25(5) (2023), 767. DOI 10.3390/e25050767. Abstract https://doi.org/10.3390/e25050767
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Computable entropy bound evaluated at approximately 0.38234.