Van der Waerden's Theorem states that for any positive integers r and k, there exists a number N such that whenever the integers from 1 to N are colored using r colors, some single color class must contain an arithmetic progression of length k. Proved by Bartel van der Waerden, it is one of the founding results of Ramsey theory, guaranteeing that arithmetic structure cannot be entirely destroyed by any finite coloring.
Facts
StatementVan der Waerden's theorem states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color. 2 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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1. Wikipedia: Van der Waerden's theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
Van der Waerden's theorem states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color.
2. Van der Waerden's Theorem (Wikipedia)
Wikimedia FoundationLead paragraph, first sentence
Van der Waerden's theorem states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color.
Lead paragraph, history sentence
This was conjectured by Pierre Joseph Henry Baudet in 1921. Waerden heard of it in 1926 and published his proof in 1927, titled Beweis einer Baudetschen Vermutung [Proof of Baudet's conjecture].
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