Frankl's union-closed sets conjecture.

Where it stands

Verified for families over ground sets up to a moderate size and for several structural special cases. The averaged and fractional relaxations have partial results, and information-theoretic approaches produced a bound bounded away from the conjectured one-half before recent improvements narrowed the gap. The general statement remains unproven.

This entry is the Atlas node; the live Solonic research campaign against it — with its own proof arc, theorems, and Ledger — is tracked separately and linked below under neighbors.

Neighbors in the graph

Every edge is typed and directed. This is what makes the Atlas a graph and not a list — and one of these crosses a domain boundary.

• Generalizes
  Lattice-theoretic union-closed variants p-math-comb-0012 · combinatorics
• Depends on
  Averaging and local-configuration bounds p-math-comb-0004 · combinatorics
• Supports
  Entropy-method lower bounds p-math-comb-0009 · within mathematics — not a bridge
• Tracked by
  Solonic FUNC campaign — Transversal Theorem, Coupling Law house research · Ledger

Revision history

The node is versioned. Every change to statement, credence, or edges is dated and kept — the Atlas has an errata trail like everything Solonic ships.

1. 2026-06-25edge added — cross-domain bridge to entropy-method bounds (p-sci-info-0002), one of 5 in the merged graph.
2. 2026-07-08erratum — a numeric credence (0.75) was displayed on this node in draft. The Atlas records no numeric credences; the field was removed rather than back-filled.
3. 2026-05-30node created — imported with statement, exit condition, and why-open axis.

Sources

1. Frankl, P. (1979). Original formulation of the union-closed conjecture.
2. Recent entropy-method improvements narrowing the constant toward one-half (2022–).
3. Solonic FUNC campaign working notes — Transversal Theorem, Deficit Law, Coupling Law (house corpus, tagged).