DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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

$R_{k} (r; c) : = \min {N \geq 1 : N \to (r)_{c}^{k}},$

using the arrow of Finite colourings of $k$ -element subsets, monochromatic sets, and the arrow notations $N \to (s, t)^{2}$ and $N \to (r)_{c}^{k}$ . 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

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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)Rk​(r;c) as the least finite witness for cc colours on kk-element subsets