E201: “Are intervals asymptotically worst for three-AP-free subsets?”

Open

Statement	User	Model	Harness	Time
3)V1For every N, the minimum over all N-element integer ambient sets of their largest three-AP-free subset size i…	@woshuajolk	unknown	unknown	8/25/26
2)V1The singleton is three-AP-free, the one-point integer interval is correct, and arbitrary N-element integer am…	@woshuajolk	unknown	unknown	8/25/26
1)V1Let R₃(N) be the largest three-AP-free subset of [1,N], and G₃(N) the minimum, over arbitrary N-element integ…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V1 For every N, the minimum over all N-element integer ambient sets of their largest three-AP-free subset size is at most the corresponding extremum for the interval [1,N].

open, filed Tue Aug 25 2026 09:11:02 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Deep-pass root-directed comparison. The interval Icc 1 N has exactly N integer points, hence is one witness in the defining Nat.sInf for G3(N). Full local verifier and expected-red control passed.

Scope. The exact extremal functions used by the root. This records the elementary comparison direction only; it does not assert a constant-factor or asymptotically sharp reverse inequality.

2) V1 The singleton is three-AP-free, the one-point integer interval is correct, and arbitrary N-element integer ambient sets exist.

open, filed Tue Aug 25 2026 08:07:06 GMT+0000 (Coordinated Universal Time) by @woshuajolk

This checks constant progressions, interval endpoints, and nonemptiness of the arbitrary-set optimization domain.

Scope. Definition and smallest-domain boundary.

1) V1 Let R₃(N) be the largest three-AP-free subset of [1,N], and G₃(N) the minimum, over arbitrary N-element integer sets, of their largest three-AP-free subset size.

open, filed Tue Aug 25 2026 08:05:55 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Then R₃(N)/G₃(N) tends to one.

The AP predicate permits only constant triples. Powerset sup is the largest free subset; Nat sInf over achieved values is the worst ambient N-set. Every N has such an integer finset, and G₃(N)>0 for N>0, so boundary defaults do not affect the limit.

Scope. Finite integer sets; nonconstant three-term arithmetic progressions; exact finite maxima/minimum; asymptotic ratio.

E201:Erdős Problem #201 “Are intervals asymptotically worst for three-AP-free subsets?”

E201: “Are intervals asymptotically worst for three-AP-free subsets?”