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Four-uniform case of Morris’s conjecture: FC(4,n)=Θ(n2)\mathrm{FC}(4,n)=\Theta(n^2)

Morris conjectured that, for each fixed k≥2k\geq 2, the Frankl-complete threshold satisfies FC(k,n)=Θ(nk−2)\mathrm{FC}(k,n)=\Theta(n^{k-2}) as n→∞n\to\infty. Here FC(k,n)\mathrm{FC}(k,n) is the least mm such that every collection of mm distinct kk-subsets of an nn-element set is Frankl-complete. A configuration G\mathcal G with support U=⋃GU=\bigcup\mathcal G is Frankl-complete if every finite union-closed family F⊇G\mathcal F\supseteq\mathcal G has an element of UU in at least half its members. This entry concerns the four-uniform case: is FC(4,n)=Θ(n2)\mathrm{FC}(4,n)=\Theta(n^2)?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI co-developed
Method
Argument
Field
Extremal set theory; union-closed families
Posed by
Robert Morris (2006), “FC-families and improved bounds for Frankl’s conjecture,” European J. Combin. 27, 269–282; Conjecture 2 in arXiv:math/0702348v1.
Year posed
2006
Years open
20y
Solved
2026-09
Model
GPT-6 Astra; Fable 5.1
Vendor
OpenAI; Anthropic
Collaborators
Mingchang Liu
Verification
Unreviewed
Publication
Preprint
Significance
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

It is proved that FC(4,n)=Θ(n2)\mathrm{FC}(4,n)=\Theta(n^2), settling the four-uniform case of Morris's conjecture; the three-uniform case had been settled exactly by Pulaj, so this was the first open uniformity. Explicitly, for n≥20n\ge 20 and r=⌊n/6⌋r=\lfloor n/6\rfloor, r(n−3r)+1≤FC(4,n)≤1+⌊1603n(n−1)⌋r(n-3r)+1\le\mathrm{FC}(4,n)\le 1+\lfloor\tfrac{160}{3}n(n-1)\rfloor, and with the Chung-Frankl triple-star theorem FC(4,n)≤(18+o(1))n2\mathrm{FC}(4,n)\le(18+o(1))n^2. The key step is a sharp sunflower criterion: four-sets with a common two-point core and pairwise disjoint two-point petals form a Frankl-complete configuration if and only if there are at least nine petals, the positive direction by a charging inequality with weight three on the core and one on the petals. Separately, exact certificates establish FC(4,9)=16\mathrm{FC}(4,9)=16 and refute Pulaj and Wood's lexicographic extremality conjecture. The paper also explains why higher-uniformity sunflowers cannot supply the same local forcing, so the general conjecture stays open.

What the AI did

The manuscript's AI statement, in full: "The author and AI models both made substantial mathematical contributions to this work. The finite classification establishing FC(4,9)=16\mathrm{FC}(4,9)=16 was developed primarily by the models. The author proposed extending this work to the four-set case of Morris's conjecture and contributed to proof development and refinement. The collaboration developed the charging inequality, 3:13:1 core-petal weights, and matching negative certificates for the sharp nine-petal sunflower threshold, and combined this local criterion with extremal bounds to prove FC(4,n)=Θ(n2)\mathrm{FC}(4,n)=\Theta(n^2). GPT-6 Astra (OpenAI) and Fable 5.1 (Anthropic) contributed to proof exploration, computation, and review. The author takes full responsibility for all mathematical results and the contents of this manuscript." Co-developed on that account: the models carried the finite classification and shared the main argument, the author set the target and takes responsibility.

Verification

Zenodo preprint V1, published 11 September 2026, one author, no independent endorsement. Checked here: the record exists with the abstract as submitted; the manuscript states Morris's Conjecture 2 as the entry does and records that Pulaj settled the three-uniform case exactly (FC(3,n)=⌊n/2⌋+1\mathrm{FC}(3,n)=\lfloor n/2\rfloor+1 for n≥4n\ge 4), so this is the first open uniformity and not a duplicate of anything in the catalog; the companion repository release V1 holds the exact certificates for FC(4,9)=16\mathrm{FC}(4,9)=16 and for the Pulaj-Wood refutation with Python and C++ verifiers. None of the mathematics was checked here and the verifiers were not run. Filed as Candidate, as submitted, until an independent reader has been through the argument or the finite certificates have been replayed.

Sources

Submitted by SilentIbis759 on

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