Morris’s union-closed-family thresholds

A finite family is Frankl-complete if every union-closed extension has an element of its original support in at least half its sets. FC(k,n) is the least m such that any m distinct k-subsets of [n] generate a Frankl-complete family. The targets are FC(4,9)=16 and FC(4,n)=Θ(n²), with a separate generalization FC(k,n)=Θ_k(n^(k−2)) for fixed k≥2.

Why it matters and what counts as progress

Why it matters: It asks for exact small thresholds and sharp asymptotic growth in a union-closed-family problem, while separating the fixed-four and general-uniformity arguments.

What counts as progress: Exact FC thresholds or sharp asymptotic orders for fixed uniformity, with the general-k extension distinguished from the four-uniform argument.

How the problem developed

2006 · Conjecture [1][2]
Robert Morris Posed the asymptotic threshold question for FC(k,n), conjecturing FC(k,n) = Θ_k(n^(k−2)) for each fixed k ≥ 2.
11 Sept 2026 · Partial result [3]
Mingchang Liu, GPT-6 Astra and Fable 5.1 Reported the four-uniform asymptotic case FC(4,n) = Θ(n²), with exact classification FC(4,9) = 16 and sunflower thresholds; the finite classification was developed primarily by the models.
13 Sept 2026 · Proof [4]
Mingchang Liu, GPT-6 Astra and Fable 5.1 Reported the all-uniformity result FC(k,n) = Θ_k(n^(k−2)) for every fixed k ≥ 2, extending the four-set result to the full conjecture.

AI contribution

Frankl-complete sunflowers and extremal families of four-sets

Mingchang Liu

Liu’s first release gives FC(4,9)=16 and a co-developed 3:1 core-petal charging argument yielding the nine-petal threshold and FC(4,n)=Θ(n²). A later version extends the asymptotic to FC(k,n)=Θₖ(n^(k−2)) for each fixed k≥2; that general proof is separate from the model-attributed first-release work.

AI involvementAI-centered

ImportanceNotable

Technical verificationPartial

Human understandingFull