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Extremal set theory · finite lattices · union-closed families

Union-Closed Sets (Frankl) Conjecture

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Must every finite nontrivial union-closed family contain an element that belongs to at least half of its sets?

maxxdF(x)≥|F|2
Known results and sources
Two input set trays feed a union tray, while an auxiliary row below shows five distinct ground-set token types without repeating the teal token; the frequency question remains unresolved.
The shared teal element appears once in the union, illustrating set deduplication; Frankl’s conjecture asks whether some element must occur in at least half the members of every finite nontrivial union-closed family.

Research problem

Exact mathematical statement

A finite family F\mathcal F of sets is union-closed if

A,B∈F⇒A∪B∈F.A,B\in\mathcal F\quad\Longrightarrow\quad A\cup B\in\mathcal F.

Frankl’s conjecture says that every finite nontrivial union-closed family contains a ground-set element belonging to at least half of its members. Writing

dF(x)=|{A∈F:x∈A}|,d_{\mathcal F}(x)=|\{A\in\mathcal F:x\in A\}|,

the desired conclusion is

maxxdF(x)≥|F|2.\max_x d_{\mathcal F}(x)\ge\frac{|\mathcal F|}{2}.

The governing source explicitly states that the full conjecture is not proved.

Problem infographic

Problem at a glance

Subset cards demonstrate union closure beside an exact eight-row occurrence table for five ground-set elements, with reconstructible frequencies 5, 7, 5, 2, and 2 and a halfway threshold of 4.
The exact occurrence table makes the example's frequency count auditable: one element appears in seven of eight sets, while the general conjecture remains open.

Current mathematical picture

Where work on Union-Closed Sets (Frankl) Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failurePromotion of incomplete or incorrectly scoped finite ledgers

Revision 8.1 records that the Turn 19 s=10 scripts, screened-set identities, task definitions, and raw terminal records are missing, so its elimination remains pending reconstruction. The serialized s=9 catalogs and exact-cross contract support a conservative exhaustive model split, followed by a search for a parameterized inequality that avoids indefinite one-layer-at-a-time enumeration.

Route status · Narrowed route
Main reductionConditional closure of one d=30 branch

The source reports 270 exact-infeasibility tasks closing (30,11,13), conditional on inherited ledger assumptions and independent model and solver audit.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeComplete the exact-cross analysis of the one-lift (30,12,14,9) layer.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Union-Closed Sets (Frankl) Conjecture in numbers

5.9kretained lines of mathematical investigation5,892 in the current working snapshot
Argument development
4,830 · 82%
Explored or eliminated routes
172 · 3%
Computational analysis
197 · 3%
Open obligations
365 · 6%
Definitions and setup
328 · 6%
6selected mapped statements1routes investigated4open questions4contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byPeter FranklProposed1979Status checkedAug 15, 2026
Source recordedAug 15, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

11 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

11 selected steps

Scroll horizontally to explore the route

Working route overview for Union-Closed Sets (Frankl) ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Some element should occur in at least half the sets — Depends on missing premiseSome element should occur inat least half the setsConditional closure of one d=30 branch — Depends on missing premiseConditional closure of oned=30 branchCurrent reduction — Depends on missing premiseCurrent reductionExact half-frequency statement — Depends on missing premiseExact half-frequencystatementFinite-lattice equivalence — Depends on missing premiseFinite-lattice equivalenceClosing target — Depends on missing premiseClosing targetPromotion of incomplete or incorrectly scoped finite ledgers — stoppedPromotion of incomplete orincorrectly scoped finiteledgersComplete the exact-cross analysis of the one-lift (30,12,14,9) layer. — OpenComplete the exact-crossanalysis of the one-lift(30,12,14,9)…Reconstruct the reported no-lift (30,12,14,10) elimination. — OpenReconstruct the reportedno-lift (30,12,14,10)elimination.Replace layer-by-layer computation with a parameterized inequality where possible. — OpenReplace layer-by-layercomputation with aparameterized…Exhaustive finite-evidence discipline — OpenExhaustive finite-evidencediscipline
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routePromotion of incomplete or incorrectly scoped finite ledgers

Revision 8.1 records that the Turn 19 s=10 scripts, screened-set identities, task definitions, and raw terminal records are missing, so its elimination remains pending reconstruction. The serialized s=9 catalogs and exact-cross contract support a conservative exhaustive model split, followed by a search for a parameterized inequality that avoids indefinite one-layer-at-a-time enumeration.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Complete the exact-cross analysis of the one-lift (30,12,14,9) layer.Suggested move: Use the source’s three incidence cases and serialized catalogs, preserve the exact nested-target two-state behavior, and record an exhaustive disjoint task cover before drawing any exclusion.
Ready to work on
02
Reconstruct the reported no-lift (30,12,14,10) elimination.Suggested move: Recover or independently rebuild the missing common-family classification, screened orbit identities, task definitions, and raw exact terminal records specified by the gap ledger.
Ready to work on
03
Exhaustive finite-evidence discipline

Any future finite exclusion must prove exhaustive coverage and classify every feasible, infeasible, timeout, unknown, numerical-error, and process-error outcome.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Replace layer-by-layer computation with a parameterized inequality where possible.Suggested move: Seek a general argument coupling weighted target counts, side content, common incidence, and coordinate capacities, while retaining safe relaxation direction.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026
Current statusOpen conjecture

Frankl's universal one-half frequency conjecture remains open. Rigorous dimension-free results guarantee an element in a fraction slightly above 0.382 of the sets, which is substantial progress but does not settle the exact target.

[3][4]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedAlweiss, Huang, and Sellke published the explicit lower bound (3−√5)/2 for the maximum element frequency.[3]
  2. Peer reviewedYu and Cambie derived computable dimension-free optimization bounds and evaluated a rigorous improvement near 0.38234.[4]
  3. PreprintGilmer proved the first absolute constant lower bound, guaranteeing an element in at least 0.01 of the sets.[2]
  4. Historical sourceThe conjecture is attributed to Peter Frankl in 1979.[1]
4 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusFrankl's union-closed sets conjecture
Weaker or relaxed formGilmer's dimension-free constant bound

A universal 0.01 frequency bound was the first dimension-free constant result but is far below the conjectured one-half threshold.

[2]
Weaker or relaxed formDimension-free approximate Frankl bounds

Modern entropy and coupling methods guarantee a frequency slightly above 0.382; they do not reach one-half.

[3][4]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • computation · source linked; not reproduced by ProofAtlasExplicit inequality check in the 0.381966 bound

    The published proof records that one explicit one-variable inequality is checked computationally; this supports the weaker constant bound, not the full conjecture.

    [3]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetNo statement-aligned formal proof of the universal one-half conjecture was identified.
  • Formalization targetThe modern entropy and coupling results would require substantial probability and information-theory infrastructure for end-to-end formalization.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

4 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role6 selected mapped statements
  • theorem candidate1 of 61
  • reduction2 of 62
  • lemma1 of 61
  • equivalence2 of 62
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included
  • retained route statementSome element should occur in at least half the sets
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact half-frequency statementintermediate
  • retained route statementFinite-lattice equivalenceintermediate
  • retained route statementConditional closure of one d=30 branchintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failurePromotion of incomplete or incorrectly scoped finite ledgersreported failure
  • Research targetComplete the exact-cross analysis of the one-lift (30,12,14,9) layer.open
  • Research targetReconstruct the reported no-lift (30,12,14,10) elimination.open
  • Research targetReplace layer-by-layer computation with a parameterized inequality where possible.open
  • Research targets=10 reconstruction gapsuperseded
  • Research targets=9 exact-cross frontiersuperseded
  • Research targetExhaustive finite-evidence disciplineopen
  • Narrowed routePromotion of incomplete or incorrectly scoped finite ledgersRevision 8.1 records that the Turn 19 s=10 scripts, screened-set identities, task definitions, and raw terminal records are missing, so its elimination remains pending reconstruction. The serialized s=9 catalogs and exact-cross contract support a conservative exhaustive model split, followed by a search for a parameterized inequality that avoids indefinite one-layer-at-a-time enumeration.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeComplete the exact-cross analysis of the one-lift (30,12,14,9) layer.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointComplete the exact-cross analysis of the one-lift (30,12,14,9) layer.

Union-Closed Sets (Frankl) Conjecture · ready to start

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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Must every finite nontrivial union-closed family contain an element that belongs to at least half of its sets?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Union-closed familiespeer reviewed result · David Duffus, Peter Frankl, Vojtěch Rödl · Journal of Combinatorial Theory, Series A · 1992 · DOI 10.1016/0097-3165(92)90068-6 · accessed Aug 15, 2026
  2. 2
    A constant lower bound for the union-closed sets conjecturepreprint · Justin Gilmer · arXiv · 2022-11-16 · ARXIV 2211.09055 · accessed Aug 15, 2026
  3. 3
    Improved Lower Bound for Frankl's Union-Closed Sets Conjecturepeer reviewed result · Ryan Alweiss, Brice Huang, Mark Sellke · The Electronic Journal of Combinatorics · 2024-09-20 · ARXIV 2211.11731 · DOI 10.37236/12232 · accessed Aug 15, 2026
  4. 4
    Dimension-Free Bounds for the Union-Closed Sets Conjecturepeer reviewed result · Lei Yu, Stijn Cambie · Entropy · 2023 · ARXIV 2212.00658 · DOI 10.3390/e25050767 · accessed Aug 15, 2026

Important qualifications

  • Scoped to primary papers for the 1979 attribution and the modern dimension-free lower-bound program; special-case literature was not exhaustively surveyed.
  • The universal one-half statement is kept distinct from lower bounds near 0.382 and from results for restricted families.
  • A 2024 arXiv manuscript claiming a complete proof was not treated as established resolution.

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