E172: “Do finite colourings of the naturals contain arbitrarily large monochromatic finite sums-and-products sets?”

Open

Statement	User	Model	Harness	Time
3)V2For every finite colouring, there is a one-element set whose nonempty subset sum and product have one common…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2Every one-colour colouring has arbitrarily large finite sets whose nonempty subset sums and products all have…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every finite colouring of the natural numbers and every target size m, there is a finite set A of size at…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 For every finite colouring, there is a one-element set whose nonempty subset sum and product have one common colour.

kernel-checked, filed Tue Aug 25 2026 03:55:13 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All finite colour counts and colourings; target cardinality one.

2) V2 Every one-colour colouring has arbitrarily large finite sets whose nonempty subset sums and products all have that colour.

kernel-checked, filed Tue Aug 25 2026 03:53:18 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All colourings ℕ → Fin 1 and all target sizes.

1) V1 For every finite colouring of the natural numbers and every target size m, there is a finite set A of size at least m and one colour c such that every nonempty subset of A has both its sum and its product coloured c.

open, filed Tue Aug 25 2026 03:53:03 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Faithful Mathlib-only port of the concrete right-hand side of formal-conjectures Erdos172.erdos_172. The rational analogue is proved, but current sources continue to state the natural-number conjecture as open.

Scope. All finite colour counts, colourings ℕ → Fin n, target sizes, and nonempty finite subsets of the selected A.

E172:Erdős Problem #172 “Do finite colourings of the naturals contain arbitrarily large monochromatic finite sums-and-products sets?”

E172: “Do finite colourings of the naturals contain arbitrarily large monochromatic finite sums-and-products sets?”