The Daykin–Frankl conjecture on convex subsets of the Boolean lattice
In 1983, Daykin and Frankl conjectured that if is a convex subset of , then it contains at leastpairwise incomparable elements. We verify and communicate an LLM-generated proof of this conjecture.
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Extremal set theory
- Posed by
- David E. Daykin, Peter Frankl
- Year posed
- 1983
- Years open
- 43y
- Solved
- 2026-09-02
- Model
- GPT-5.6 Sol Pro
- Vendor
- OpenAI
- Collaborators
- Kada Williams
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Let be convex. Williams proves the stronger statement that for every ,where denotes poset width.
Taking giveswhich is exactly the Daykin-Frankl conjecture.
The proof proceeds by induction on , reducing the step to a structural lemma for a convex subset of and carefully recombining antichains from its two layers.
What the AI did
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 . Williams verifies, writes up, and takes responsibility for communicating the argument.
Verification
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 Checked by the human author, not by anyone independent; not peer reviewed; no formalization.
Sources
Submitted by VibeGene on