problems / union-closed

Also in this folder: compute/ · figures/

Frankl's union-closed sets conjecture

Statement

Every finite nontrivial union-closed family is conjectured to have an element in at least half its sets. Gilmer gave a positive constant; the 1/2 threshold is open.

Tonight

A certified improvement of the published frequency constant, or a new finite classification with a verifier. Do not claim the 1/2 conjecture unless you prove it. Fetch Gilmer and the current best constant before searching.

Result (2026-08-27, q3)

Dent: pure Liu Example 4 on {b,1} certifies 0.38305 > 0.38304 > 0.382709087918741. The last number is Liu's published Example 5 value. The analytic first-crossing remains 0.38305135658682558…, leaving 1.3565868×10⁻⁶ above the printed constant. On a 9,000×7,000 mesh, every retained cell with mean at most 0.38305 has Gilmer ratio greater than 1. Python row-boundary and exhaustive C verifiers agree on 20,440,358 retained cells, zero bad cells, and minimum ratio 1.0000049143029008. Replay: compute/q3/run_all.sh (exit 0). A shared-input joint-entropy probe reaches equality near 0.343708. This leaves residue, and the q2 ceiling is unchanged. The scope remains the {b,1} family; Frankl's 1/2 target and the every-measure inequality are open.

Result (2026-08-27, q2)

The constant did not move. On {b,1}, every 2-sample bit protocol has first-crossing at most the q1 critical point 0.3830513565868…, because h(Π_{b,b}) ≤ 1. Mixes (new β, Example 5, max-entropy), a half-target Fréchet protocol, and scaled a(t) all sit at or below that ceiling. Gilmer's KL path to 1/2 is Ellis's n=2 counterexample, replayed. Pure Example 4 fails a constructed 2-mixture at mean 0.38304 (CIID ratio 0.909137); that is residue off the ray, not a dent. Replay: compute/q2/run_all.sh (exit 0). Still not 1/2.

Result (2026-08-27)

On the two-point family {b,1}, pure Liu Example 4 (β = 1) has first-crossing equal to the unique critical point of 1 − (1−b)h(b) on (1−1/√2, 1/2]. That number is 0.3830513565868…, defined by h(b) = (1−b) log₂((1−b)/b). The certified 5-decimal constant is 0.38304: every mesh cell with mean ≤ 0.38304 has Gilmer ratio ≥ 1.0000217 (Python and C agree). This beats the 2026-08-17 ray number 0.38285 (β = 1/5) and Liu's published 0.382709. Replay: compute/q1/run_all.sh (exit 0). Same hypothesis class as Liu Theorem 13. Not 1/2. Not every measure on [0,1].

Result (2026-08-17)

On the two-point family {b,1} that Liu identified as the optimizer, the iid + Example-4 mix with weight β = 1/5 has Gilmer ratio ≥ 1 for every mesh cell with mean ≤ 0.38285. Liu's published number on the same family, using Example 5, is 0.382709. Replay: compute/verify.py (exit 0). This is not a proof of the 1/2 conjecture, and it is not an unconditional theorem for every measure — it is the same hypothesis class as Liu Theorem 13, with a better protocol and an independent mesh.