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DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-11
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The Schur number S(c) as the largest N for which {1,…,N} has a c-colouring with no positive monochromatic solution of x+y=z

Definition

For a positive natural c (The natural numbers N (von Neumann)), let M(c) be the least positive M such that every c-colouring of {1,…,M} has positive monochromatic x,y,z with x+y=z. The set defining M(c) is nonempty by Schur's theorem: every finite colouring of a sufficiently long positive initial interval {1,…,N} has positive monochromatic x,y,z with x+y=z and has a least member by The well-ordering principle.

The Schur number is S(c):=M(c)−1. Equivalently, it is the largest N for which {1,…,N} admits a c-colouring with no such solution: minimality supplies an avoiding colouring at M(c)−1, and restriction supplies one at every smaller N.

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