DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-11

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The van der Waerden number $W(k,c)$ as the least interval length forcing a monochromatic $k$ -term arithmetic progression

Definition

For positive naturals $k,c$ (The natural numbers $\mathbb{N}$ (von Neumann)), the van der Waerden number $W(k,c)$ is the least positive $N$ such that every $c$ -colouring of any interval of $N$ consecutive integers contains a monochromatic $k$ -term arithmetic progression with positive common difference.

Such an $N$ exists by Van der Waerden's theorem, strengthened so the progression and its common difference have one colour, whose stronger conclusion also colours the common difference, and leastness follows from The well-ordering principle. Translation identifies every interval of $N$ consecutive integers with $\{1,\ldots,N\}$ without changing arithmetic progressions.

Depends on

Used by

W(3,2)=9 by an explicit colouring of {0,…,7} and an exhaustive symmetry-reduced proof for {0,…,8} Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 32 results over 13 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, Section 1.2 (standard reference, not scraped)

The van der Waerden number W(k,c)W(k,c)W(k,c) as the least interval length forcing a monochromatic kkk-term arithmetic progression