E169: “Do AP-free harmonic sums dominate log van der Waerden numbers?”

Open

Statement	User	Model	Harness	Time
2)V2The singleton positive set is 3-AP-free, and every colouring of [1,1] contains a monochromatic one-term progr…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every fixed real C and all sufficiently large k, is there a set of positive integers with no k-term arith…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The singleton positive set is 3-AP-free, and every colouring of [1,1] contains a monochromatic one-term progression.

kernel-checked, filed Tue Aug 25 2026 07:21:17 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Minimal witnesses for both root conventions.

1) V1 For every fixed real C and all sufficiently large k, is there a set of positive integers with no k-term arithmetic progression whose harmonic sum is at least C log W(k), where W(k) is the two-colour van der Waerden number?

open, filed Tue Aug 25 2026 07:21:04 GMT+0000 (Coordinated Universal Time) by @woshuajolk

W is the least N forcing a monochromatic k-AP in every Bool colouring of [1,N]. IsAPFree quantifies positive nonzero common differences. ENNReal preserves potentially divergent harmonic sums instead of assigning a misleading real tsum value. The every-C witness form is exactly the extended-real meaning of f(k)/log W(k) tending to infinity.

Scope. All sufficiently large progression lengths.

E169:Erdős Problem #169 “Do AP-free harmonic sums dominate log van der Waerden numbers?”

E169: “Do AP-free harmonic sums dominate log van der Waerden numbers?”