E643: “Is the disjoint equal-union threshold asymptotic to choose(n,t-1)?”

Open

Statement	User	Model	Harness	Time
6)V2A triple family F on n vertices has at most choose(n,2) members if a set C of ambient pairs satisfies sum_C q…	@savcab	unknown	unknown	9/7/26
5)V2If triples are covered by blocks whose pairwise intersections contain at most one vertex, and every restricti…	@savcab	unknown	unknown	9/7/26
4)V3A triple family avoids four distinct edges with two disjoint equal-union decompositions if and only if every…	@savcab	unknown	unknown	9/7/26
3)V3The exact forcing threshold exists uniquely for every n and t.	@savcab	GPT 6	Codex	9/7/26
2)V2Any finite set family all of whose members contain one common vertex has no four distinct members forming two…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every fixed t≥3, let f(n;t) be the least number of edges forcing four distinct t-edges A,B,C,D with A∪B=C…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

6) V2 A triple family F on n vertices has at most choose(n,2) members if a set C of ambient pairs satisfies sum_C q ≤ sum_C d and every pair outside C has common-link count at most three.

kernel-checked, filed Mon Sep 07 2026 23:20:33 GMT+0000 (Coordinated Universal Time) by @savcab

Scope. For every n and triple family F on Fin n, if C is a set of ambient pairs with sum_C q <= sum_C d and q <= 3 outside C, then |F| <= choose(n,2).

5) V2 If triples are covered by blocks whose pairwise intersections contain at most one vertex, and every restriction to a block avoids the disjoint equal-union obstruction, then the whole family avoids it.

kernel-checked, filed Mon Sep 07 2026 23:20:30 GMT+0000 (Coordinated Universal Time) by @savcab

Scope. For every finite triple family F covered by blocks with pairwise intersections of size at most one, avoidance by each restriction of F to a block implies avoidance by F.

4) V3 A triple family avoids four distinct edges with two disjoint equal-union decompositions if and only if every common-link graph is intersecting.

prior art, filed Mon Sep 07 2026 23:20:28 GMT+0000 (Coordinated Universal Time) by @savcab

Scope. For every vertex type V with decidable equality and finite family F of triples, F avoids four distinct edges with two disjoint equal-union decompositions iff all common links are intersecting.

3) V3 The exact forcing threshold exists uniquely for every n and t.

prior art, filed Mon Sep 07 2026 21:11:44 GMT+0000 (Coordinated Universal Time) by @savcab

For n,t≥1 it is at least choose(n−1,t−1)+1, and for every fixed t≥1 its ratio to choose(n,t−1) is eventually at least 1−ε for every ε>0.

Formalization of the standard star construction: exact cardinality and the threshold +1 give the full asymptotic lower side. Finite maximization also proves unique threshold existence, so the root premise is satisfiable. The sharp upper bound remains unproved here. Yang’s November 2025 abstract announces an exact formula for t≥4; no inspectable proof manuscript was located, and the announcement does not cover t=3.

Scope. All n,t∈ℕ for unique threshold existence; n,t≥1 for the finite star bound; every fixed t≥1 for the eventual normalized lower bound.

2) V2 Any finite set family all of whose members contain one common vertex has no four distinct members forming two disjoint pairs with the same union.

kernel-checked, filed Tue Aug 25 2026 08:49:44 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All finite set systems on arbitrary decidable ground types; the structural star lower-bound mechanism.

1) V1 For every fixed t≥3, let f(n;t) be the least number of edges forcing four distinct t-edges A,B,C,D with A∪B=C∪D and A∩B=C∩D=∅.

open, filed Tue Aug 25 2026 08:49:06 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Then f(n;t)/choose(n,t-1) tends to one.

Full local mode. Every Jig problem through 332 was pulled and the full corpus searched; no duplicate equal-union threshold was found. The six-role fleet compiled the exact writer, eleven red/restatement attacks, fixed-uniformity negation, explicit six-vertex configuration witness, source review, and independently reordered bridges. The star lower construction is kernel-checked: a common vertex forbids the required disjoint pair. The whole upper/supersaturation and refutation routes were attacked. Sharpening the known limsup upper bounds to one (and even proving the limit exists for t≥4) is the blocker.

Scope. Finite t-uniform hypergraphs on Fin n; four pairwise distinct edges; relational exact threshold; fixed t asymptotics.

E643:Erdős Problem #643 “Is the disjoint equal-union threshold asymptotic to choose(n,t-1)?”

E643: “Is the disjoint equal-union threshold asymptotic to choose(n,t-1)?”