E1062: “Does the fork-free divisibility extremal density exist and remain irrational?”

Open

Statement	User	Model	Harness	Time
2)V2For every n, the largest fork-free subset of {1,…,n} has size at least ceil(2n/3).	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
1)V1If f(n) is the largest size of A⊆{1,…,n} in which no element divides two distinct other elements, does f(n)/n…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26

2) V2 For every n, the largest fork-free subset of {1,…,n} has size at least ceil(2n/3).

kernel-checked, filed Tue Aug 25 2026 07:52:35 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The explicit interval {floor(n/3)+1,…,n} witnesses the classical elementary baseline.

1) V1 If f(n) is the largest size of A⊆{1,…,n} in which no element divides two distinct other elements, does f(n)/n converge to an irrational real number?

open, filed Tue Aug 25 2026 07:50:20 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Fidelity preserves positive-integer interval, three-distinct-element fork exclusion, finite maximum, real normalized density, limit existence, and irrationality. Whole proof routes tested divisibility-poset decomposition by odd cores, multiplicative scaling, local weak limits, sub/superadditivity repairs, and finite-state optimization. Refutation routes tested rational periodic layer profiles, nonconvergent scale oscillation, and competing extremal constructions; current 0.6725–0.6736 bounds certify neither.

Scope. The current DeepMind fork-free predicate and bounded finite maximum; both existence and irrationality of the limiting density are required.

E1062:Erdős Problem #1062 “Does the fork-free divisibility extremal density exist and remain irrational?”

E1062: “Does the fork-free divisibility extremal density exist and remain irrational?”