Research log — Frankl's union-closed sets conjecture

2026-08-17

Fetched and read tonight, in the order used.

Primary (frequency constant)

• Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055v2 (28 Nov 2022). First constant 0.01. Entropy of two iid samples. Conjectures the iid limit φ=(3−√5)/2. Local PDF: compute/refs/gilmer-2211.09055.pdf.
• Alweiss–Huang–Sellke, Improved lower bound…, arXiv:2211.11731; EJC 31(3) P35 (2024). Proves Gilmer's φ. One 1-variable inequality checked by computer. Local: compute/refs/alweiss-2211.11731.pdf.
• Chase–Lovett, Approximate union-closed conjecture, arXiv:2211.11689. φ is sharp for approximate union-closed families. Local: compute/refs/chase-lovett-2211.11689.pdf.
• Sawin, An improved lower bound…, arXiv:2211.11504v3 (19 Jun 2023). Mix iid with max-entropy coupling; proves some c>φ without evaluating it. Local: compute/refs/sawin-2211.11504.pdf.
• Yu, Dimension-free bounds…, arXiv:2212.00658v2; Entropy 25(5):767 (2023). Makes Sawin computable; c*≈0.38234. Cardinality reduction to two 2-point blocks. Local: compute/refs/yu-2212.00658.pdf.
• Cambie, Better bounds…, arXiv:2212.12500v2 (16 Feb 2025). Evaluates Sawin exactly: 0.382345533366702 ≤ c* ≤ 0.382345533366703, α*≈0.03560698136437784. Code: https://github.com/StijnCambie/UCconjecture. Calls the computer check “slightly less rigorous”. Local: compute/refs/cambie-2212.12500.pdf.
• Liu, Improving the lower bound… via conditionally IID coupling, arXiv:2306.08824 (15 Jun 2023). Theorem 6: some unspecified c>c* (Example 4 perturbation). Theorem 13: c′≈0.382709 under PSD + 9-D global-min hypotheses, Example 5. Local: compute/refs/liu-2306.08824.pdf.

Status quotes (what the literature treats as “the” constant)

• Lu–Raz, Note on the union-closed sets conjecture and Reimer's average set size theorem, arXiv:2405.10639v2 (30 May 2024): “The best constant lower bound currently is approximately 0.38271, proven by Liu.” Local: compute/refs/lu-raz-2405.10639.pdf.
• Wikipedia, Union-closed sets conjecture, oldid 1350313043, fetched 2026-08-17: quotes 0.38271 and cites Liu + Lu–Raz.
• Das–Janzer–Sudakov, Frequent elements in union-closed set families, arXiv:2412.03862: treats Liu's number as the standing frequency bound; k-th most frequent element matches the Gilmeresque constant for large families.

Finite cases (not moved tonight)

• Bruhn–Schaudt, The journey of the union-closed sets conjecture, arXiv:1309.3297; Graphs Combin. 31 (2015). Survey through 2013.
• Roberts–Simpson, A note on the union-closed sets conjecture, Australas. J. Combin. 47 (2010) 265–267. Minimal counterexample has |A|≥4q−1. Combined with q≥13 this is |F|≥51, hence the |F|≤50 statement.
• Vučković–Živković, The 12-element case of Frankl's conjecture, IPSI BGD Trans. Internet Research 13 (2017). Universe size ≤12.
• [Sarvate–Renaud, On the union-closed sets conjecture, Ars Combin. 27 (1989)]. Smallest set of size 1 or 2 forces an abundant element of that set; size 3 does not (Graham-type examples).

False or unused full-proof claims

• Scandone, arXiv:2302.03484, “A proof of the union-closed sets conjecture”. Later literature still treats the conjecture as open.
• A January 2026 HAL note claiming an “algorithmic proof” was not used.

What we compare against

• Best unconditional explicit published constant with a complete reduction: Yu–Cambie c* = 0.3823455333667027… (computer-checked 2-variable inequality). Independently recomputed tonight, residual <10^{-15} vs Cambie's quote.
• Best quoted published constant: Liu 0.382709087918741, under two numerical hypotheses. Independently recomputed: 0.382709087918735. Mesh first-crossing of his Example-5 mix on {b,1}: 0.38271065.
• Tonight's number 0.38285 beats Liu on that same family, with his Example 4 protocol. It does not replace Yu–Cambie as an unconditional theorem, and it is not 1/2.

2026-08-27

Fetched and read tonight, in the order used. No paper after Liu moves the frequency constant.

Primary (frequency constant, re-opened)

• Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055v2 (28 Nov 2022). Re-fetched. Theorem: some element in a 0.01 fraction. Does not claim φ as a theorem; conjectures it as the iid limit. Local PDF: compute/refs/gilmer-2211.09055.pdf.
• Liu, Improving the lower bound… via conditionally IID coupling, arXiv:2306.08824v1 (15 Jun 2023). Re-fetched HTML. Example 4 is (22)–(23); Theorem 6 is existential c > c*; Theorem 13 is c′ ≈ 0.382709087918741 under PSD + global-min, Example 5, β* ≈ 0.10005. No Example-4 numerical section. Local: compute/refs/liu-2306.08824.pdf.
• Lu–Raz, Note on the union-closed sets conjecture and Reimer's average set size theorem, arXiv:2405.10639. Re-opened abs page. Still: “The best constant lower bound currently is approximately 0.38271, proven by Liu.”
• Wikipedia, Union-closed sets conjecture, fetched 2026-08-27. Still quotes 0.38271 and cites Liu + Lu–Raz. Finite cases still 50 sets / 12 elements.
• Das–Wu, Frequent elements in union-closed set families, arXiv:2412.03862v3 (11 Jul 2025). Re-opened abs. k-th frequent element; large-family frequency φ − o(1). Intro treats 0.3823455 as “where things stand today” and cites Liu's CISS writeup separately. Does not claim a number above Liu.

Later papers that do not move the frequency constant

• DeFranco, On Boolean polynomials and the Union-Closed Conjecture, arXiv:2606.26191 (24 Jun 2026). Encoding: the conjecture for given (m,n) iff a Boolean polynomial is identically zero. No numerical constant.
• Tian, Frankl's Conjecture at Height Four…, arXiv:2608.25147 (25 Aug 2026). Proves the empty-set-free form for poset height ≤ 4. Structural constraints at height 5. Not a frequency constant; first-open universe size remains 13 in the papers that state that frontier.
• Gendler, Partial results for union-closed conjectures on the weighted cube, arXiv:2504.13347v3 (4 Nov 2025). Weighted product measure; Karpas-type density and Knill log bound. No 0.38-style constant.
• arXiv API query all:union-closed AND all:conjecture, sorted by submitted date, 30 hits (2026-08-27). No hit after Liu claims a frequency constant above 0.38271.

What we compare against tonight

• Best quoted published constant: still Liu 0.382709087918741 (conditional). Independently recomputed: 0.382709087918735.
• Repo ray-record from 2026-08-17: 0.38285 at β = 1/5, mesh min ratio 1.000077. Replayed; compute/run_all.sh exit 0.
• Tonight: analytic first-crossing of pure Example 4 on {b,1} is 0.38305135658682558…; certified 0.38304 with mesh min ratio 1.000021687. Beats both. Same hypothesis class. Not 1/2.

2026-08-27 (q2)

Fetched and read tonight, in the order used. No paper after Liu moves the frequency constant. The 2-sample {b,1} class cannot pass 0.383051.

Primary (re-opened)

• Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055v2 (28 Nov 2022). Re-fetched abs + HTML. Theorem: some element in a 0.01 fraction. §5 Conjecture 1 (H(A∪B)+D(A∪B||A)>H(A) whenever every frequency is <1/2) would imply Frankl 1/2. v2 note: Sawin and Ellis refute Conjecture 1; without extra assumptions the KL term cannot improve the bound. Local PDF: compute/refs/gilmer-2211.09055.pdf.
• Liu, Improving the lower bound… via conditionally IID coupling, arXiv:2306.08824v1 (15 Jun 2023). Re-fetched abs + HTML. Theorem 6: some unspecified c>c* via Example 4. Theorem 13: c′≈0.382709087918741 under PSD + global-min, Example 5, β*≈0.10005. Example 4 is (22)–(23). No later version. Local: compute/refs/liu-2306.08824.pdf.
• Ellis, Note: a counterexample to a conjecture of Gilmer…, arXiv:2211.12401 (22 Nov 2022). Opened abs + HTML. n=2 law p(∅)=p({1,2})=0.3, p({1})=p({2})=0.2; functional < −0.04; a small perturbation has frequencies <1/2 and stays negative. Independently replayed: −0.046797 unperturbed, −0.043422 at ε=10^{-3}.
• Wikipedia, Union-closed sets conjecture, fetched 2026-08-27. Still quotes 0.38271 and cites Liu + Lu–Raz. Finite cases still 50 sets / 12 elements.

Later papers that do not move the frequency constant

• arXiv API query all:"union-closed" AND all:conjecture, submitted-date desc, 25 hits (2026-08-27). Newest frequency-adjacent items are Tian 2608.25147 (height 4), DeFranco 2606.26191 (Boolean encoding), Das–Wu 2412.03862 / 2412.03863 (k-th frequency). No hit after Liu claims a first-frequency constant above 0.38271.
• Tian, arXiv:2608.25147. Height ≤ 4. Not a frequency constant.
• “The union-closed set conjecture is true”, arXiv:2405.03731. Title-only full-proof claim; later literature and Wikipedia still treat the conjecture as open. Not used.

What we compare against tonight

• Best quoted published constant: still Liu 0.382709087918741 (conditional). Independently recomputed: 0.382709087918735.
• Repo ray-record from q1: 0.38304, analytic crossing 0.38305135658682558…. Replayed; compute/q1/run_all.sh exit 0.
• Tonight: that crossing is a ceiling for every 2-sample bit protocol on {b,1}. Mixes and a half-target protocol do not beat it. A constructed 2-mixture has pure-Example-4 CIID ratio 0.909137 at mean 0.38304. Constant unchanged. Not 1/2.

2026-08-27 (q3)

• Liu, Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling, arXiv:2306.08824v1 (15 Jun 2023). Re-opened the current arXiv record. It still has only v1, and the abstract still states ≈0.38271 under numerically verified hypotheses. The repository fetch helper was also attempted and received an arXiv API 429 response.
• Gilmer, A constant lower bound for the union-closed sets conjecture, arXiv:2211.09055v2 (28 Nov 2022). Re-opened the current record. The theorem remains the 0.01 frequency, and the v2 comments still note follow-up work refuting Conjecture 1.
• Ellis, Note: a counterexample to a conjecture of Gilmer which would imply the union-closed conjecture, arXiv:2211.12401v1 (22 Nov 2022). Re-opened the current record; the three-page counterexample remains the obstruction replayed in q2.

What we compare against

• Liu's quoted published number is 0.382709087918741, conditional in the paper's own wording.
• q1's ray-certified notebook number is 0.38304, with analytic crossing 0.38305135658682558….
• q3 certifies the next five-decimal value 0.38305 on the same {b,1} family. This is a notebook ray result; Frankl's 1/2 target and the every-measure inequality remain open.