The Daykin–Frankl conjecture on convex subsets of the Boolean lattice

Precise statement

In 1983, Daykin and Frankl conjectured that if $P$ is a convex subset of $Q_n$, then it contains at least $$ |P|\binom{n}{\lfloor n/2\rfloor}2^{-n} $$ pairwise incomparable elements. We verify and communicate an LLM-generated proof of this conjecture.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT-5.6 Sol Pro

Kada Williams explicitly credits ChatGPT 5.6 Sol Pro with generating the proof content of the note. The proof establishes a stronger product inequality for convex subsets of Boolean lattices and derives the original Daykin-Frankl conjecture as the case $k=0$. Williams verifies, writes up, and takes responsibility for communicating the argument.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

Unreviewed. arXiv 2609.03087 (four pages) read here: the note describes itself as verifying and communicating an LLM-generated proof, credits ChatGPT 5.6 Sol Pro, and gives the induction on dimension with the R x Q_1 convexity lemma in full. Checked by the human author, not by anyone independent; not peer reviewed; no formalization.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-09-02

   arXiv

   Let $P\subseteq Q_n$ be convex. Williams proves the stronger statement that for every $k\ge0$, $$ w(P\times Q_k) \ge w(Q_{n+k})\,|P|\,2^{-n}, $$ where $w$ denotes poset width. Taking $k=0$ gives $$ w(P)\ge |P|\binom{n}{\lfloor n/2\rfloor}2^{-n}, $$ which is exactly the Daykin-Frankl conjecture. The proof proceeds by induction on $n$, reducing the step to a structural lemma for a convex subset of $R\times Q_1$ and carefully recombining antichains from its two layers.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.