DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 $N$ (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 ${1, \dots, N}$ 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

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Sources

I. B. Leader, Ramsey Theory, Section 1.2 (standard reference, not scraped)

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