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

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The uniform Ramsey number $R_k(r;c)$ as the least finite witness for $c$ colours on $k$ -element subsets

Definition

For positive natural numbers $k,r,c$ , the uniform Ramsey number is

$R_k(r;c):=\min\{N\ge1:N\to(r)^k_c\},$

using the arrow of Finite colourings of $k$ -element subsets, monochromatic sets, and the arrow notations $N\to(s,t)^2$ and $N\to(r)^k_c$ . The defining set is nonempty by For positive $k,c,r$ there is an $N$ such that every $c$ -colouring of $[N]^k$ has a monochromatic $r$ -element set and therefore has a least member by The well-ordering principle.

Depends on

Used by

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Sources

R. Diestel, Graph Theory, 6th ed., Chapter 9, Section 9.1 (standard reference, not scraped)

The uniform Ramsey number Rk(r;c)R_k(r;c)Rk​(r;c) as the least finite witness for ccc colours on kkk-element subsets