False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: every two-colouring of $\mathbb N$ contains an infinite monochromatic arithmetic progression

Statement

Every two-colouring of $\mathbb N$ contains an infinite monochromatic arithmetic progression $a,a+d,a+2d,\ldots$ with $d>0$ .

Facts & Assumptions

Given: Natural numbers and their order as in The natural numbers $\mathbb{N}$ (von Neumann) and Order on the natural numbers.

[L1]

Every finite colouring of a sufficiently long initial interval has a monochromatic arithmetic progression whose common difference has the same colour (Van der Waerden's theorem, strengthened so the progression and its common difference have one colour).

Refutation

technique · constructive

1.1

Colour $0$ red. For $n\ge1$ , colour $n$ red when the unique $m$ with $2^m\le n<2^{m+1}$ is even, and blue when $m$ is odd. Thus consecutive dyadic blocks alternate colours, with every power of two assigned to the block beginning there.

construct

2.1

Fix $a\in\mathbb N$ and $d>0$ . For every sufficiently large $m$ , let $q_m$ be the least $q$ with $a+q d\ge2^m$ . Minimality gives $a+q_m d<2^m+d<2^{m+1}$ , so the progression meets the $m$ th dyadic block. It therefore meets infinitely many blocks of each parity and contains both colours.

step 1.1algebra

3.1

No infinite arithmetic progression is monochromatic in this colouring. This does not contradict [L1], which guarantees arbitrarily long finite progressions only. The displayed universal statement is false.

step 2.1L1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

I. B. Leader, Ramsey Theory, remark after Corollary 7 (standard reference, not scraped)

FALSE: every two-colouring of N\mathbb NN contains an infinite monochromatic arithmetic progression