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.
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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
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Source Erdos-Rado theorem (Wikipedia)
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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 expr(κ)+, in κ many colors, then there is a homogeneous set of cardinality κ+ (a set, all whose r+1-element subsets get the same f-value).
References, Erdos and Rado (1956) entry
Erdős, P.; Rado, R. (1956)
- In Branch: Set Theory, Lead sentence
In Group: Ramsey Theory and Extremal Combinatorics, Lead section
the Erdős-Rado theorem is a basic result extending Ramsey's theorem to uncountable sets. It is named after
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