The Erdos-Rado Theorem, a basic result of partition calculus within combinatorial set theory, extends Ramsey's Theorem from finite to uncountable sets. It is named after Paul Erdos and Richard Rado, and is sometimes attributed as well to Duro Kurepa, who proved a version of it under the additional assumption of the generalized continuum hypothesis, so the result is occasionally called the Erdos-Rado-Kurepa theorem.
Facts
Partially Attested
Proof YearYear is from the bibliography entry for Erdos and Rado (1956); the article does not state when the theorem was proved. StatementIf f is a coloring of the r+1-element subsets of a set of cardinality exp_r(kappa)^+ in kappa many colors, then there is a homogeneous set of cardinality kappa^+, for r finite and kappa an infinite cardinal. 1 Classification
Statement Form Connections
Has Statement Form
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.
In Branch
Source Erdos-Rado theorem (Wikipedia)
Proved By
Sources
1. Erdos-Rado theorem (Wikipedia)
Statement of the theorem
If f is a coloring of the r+1-element subsets of a set of cardinality exp_r(╬║)+, in ╬║ many colors, then there is a homogeneous set of cardinality ╬║+
References, Erdos and Rado (1956) entry
Erdős, P.; Rado, R. (1956)
- In Branch: Set Theory, Lead sentence
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