Maximum size of a three-sunflower-free family of four-element sets
How large can a family of distinct four-element sets be if no three of its members form a sunflower, that is, three distinct members whose pairwise intersections are all equal (an empty common intersection included)? Write for this maximum.
Abbott, Hanson and Sauer (1972) gave and Abbott and Hanson (1974) gave . Axante, Budala, Chitic, Dumitru and Nacu (arXiv:2609.06175v1, September 2026, Table 1 and Theorem 6.9) improved the upper bound to 83, so . The exact value is open. The forcing-threshold convention used for Erdős Problem #20 is one larger than .
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-assisted
- Method
- Computation
- Field
- Extremal set theory; sunflowers and finite set systems
- Posed by
- Abbott, Hanson and Sauer (1972, upper bound 142) and Abbott and Hanson (1974, lower bound 54), as recorded in Axante et al., arXiv:2609.06175v1, Table 1
- Year posed
- —
- Years open
- —
- Solved
- 2026-10
- Model
- Sol 5.6; Astra 6; Pro 6; Sol 6.1
- Vendor
- OpenAI
- Collaborators
- Oleksiy Babanskyy
- Verification
- Unreviewed
- Publication
- Announced
- Significance
- 9 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
The manuscript claims a computer-assisted exclusion of 69-member three-sunflower-free four-uniform families, giving 54 <= M(4,3) <= 68. It improves the upper bound 83 in Axante et al., arXiv:2609.06175v1, Theorem 6.9. The classical lower construction is reproduced, not claimed as new. The proof combines their intersecting-family bound 27, structural point-link reductions, Jin's graph-homomorphism theorem, exact certificates and finite exhaustive checks. A local cap of 61 requires a complete R16 point-link and is not a global bound. Cardinality 68 is not excluded, the exact value is not determined, and the general sunflower conjecture (Erdős #20) remains open.
What the AI did
The author reports using OpenAI Sol 5.6, Astra 6, Pro 6 and Sol 6.1. As disclosed in the manuscript, generative AI tools assisted proof development, software preparation and manuscript review. The named human author retains responsibility for the claims. The model names are the author's disclosure for this submission; the manuscript does not identify versions or attribute individual lemmas to particular models. Author-side AI reviews are not independent specialist verification.
Verification
No independent mathematician has checked this yet. Checked by this site on 15 October 2026: the aiXiv v1 record and PDF open (CC BY, content-hashed, not withdrawn), the pinned GitHub commit 90c03c7 exists, and the manuscript's AI disclosure and Theorem 1.1 were read. Axante et al., arXiv:2609.06175v1 (still the only version), was read for Table 1, Proposition 6.7 (the imported bound 27) and Theorem 6.9 (54 <= f(4,3) <= 83). A separate script written here confirmed that the 54-member construction has no sunflower among its 24,804 triples, that no further four-set on its 18 points can be added, and that a graph with no three-edge sunflower has at most six edges (Lemma 3.1, on seven vertices). The companion's 42 finite suites (about 53 minutes) were not re-run here, nor was the imported exhaustive computation behind the bound 27, itself an AI-assisted preprint result from September 2026. The upper bound 68 rests on those computations and the written reductions; the author-side AI reviews are not independent.
Sources
Submitted by Oleksiy Babanskyy on