Morris's asymptotic conjecture: $\mathrm{FC}(k,n)=\Theta_k(n^{k-2})$

Morris's Conjecture 2 asks whether, for every fixed integer $k\ge2$ , $\mathrm{FC}(k,n)=\Theta_k(n^{k-2})$ as $n\to\infty$ . Here $\mathrm{FC}(k,n)$ is the least $m$ such that every collection of $m$ distinct $k$ -subsets of an $n$ -point set is Frankl-complete. A configuration $\mathcal G$ with support $U=\bigcup\mathcal G$ is Frankl-complete if every finite union-closed family $\mathcal F\supseteq\mathcal G$ has an element of $U$ belonging to at least half its sets. The extension may contain additional points.

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 $\mathrm{FC}(k,n)=\Theta_k(n^{k-2})$ for every fixed $k\ge2$ , resolving Morris's Conjecture 2. The proof combines robust weighted certificates, pair lifting and extremal forcing with a self-contained lower-bound argument.

For four-sets, $n^2/4-O(n)\le\mathrm{FC}(4,n)\le(18+o(1))n^2$ . More generally, for fixed $k\ge4$ , $\mathrm{FC}(k,n)\ge\frac{n^{k-2}}{2(k-2)!}-O_k(n^{k-3})$ .

The sharp FC and robustness thresholds for sunflowers with a two-point core and disjoint pair petals are nine and ten petals. Exact finite certificates establish $\mathrm{FC}(4,9)=16$ and refute Pulaj and Wood's lexicographic extremality conjecture.

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 $\mathrm{FC}(4,9)=16$ 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, $3:1$ core-petal weights, and matching negative certificates for the sharp nine-petal sunflower threshold, and combined this local criterion with extremal bounds to prove $\mathrm{FC}(4,n)=\Theta(n^2)$ . 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 ( $\mathrm{FC}(3,n)=\lfloor n/2\rfloor+1$ for $n\ge 4$ ), 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 $\mathrm{FC}(4,9)=16$ 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

PaperZenodo preprint (V2)
CodeV1 code and exact certificates
Problem recordMorris’s original conjecture (Conjecture 2)

Submitted by SilentIbis759 on 11 Sep 2026

Morris's asymptotic conjecture: FC(k,n)=Θk(nk−2)\mathrm{FC}(k,n)=\Theta_k(n^{k-2})FC(k,n)=Θk​(nk−2)

What was actually shown

What the AI did

Verification

Sources

Discussion

Morris's asymptotic conjecture: $\mathrm{FC}(k,n)=\Theta_k(n^{k-2})$