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Inmathematics, aRamsey cardinal is a certain kind oflarge cardinal number introduced byErdős & Hajnal (1962) and named afterFrank P. Ramsey, whose theorem, calledRamsey's theorem establishes thatω enjoys a certain property that Ramsey cardinals generalize to theuncountable case.
Let [κ]<ω denote the set of all finite subsets ofκ. Acardinal numberκ is called Ramsey if, for every function
there is a setA of cardinalityκ that ishomogeneous forf. That is, for everyn, the functionf isconstant on the subsets of cardinalityn fromA. A cardinalκ is calledineffably Ramsey ifA can be chosen to be astationary subset ofκ. A cardinalκ is calledvirtually Ramsey if for every function
there isC, a closed and unbounded subset ofκ, so that for everyλ inC of uncountablecofinality, there is an unbounded subset ofλ that is homogenous forf; slightly weaker is the notion ofalmost Ramsey where homogenous sets forf are required of order typeλ, for everyλ <κ.
The existence of any of these kinds of Ramsey cardinal is sufficient to prove the existence of0#, or indeed that every set withrank less thanκ has asharp. This in turn implies the falsity of theAxiom of Constructibility ofKurt Gödel.
Everymeasurable cardinal is a Ramsey cardinal, and every Ramsey cardinal is aRowbottom cardinal.
A property intermediate in strength between Ramseyness andmeasurability is existence of aκ-complete normal non-principalidealI onκ such that for everyA ∉I and for every function
there is a setB ⊂A not inI that is homogeneous forf. This is strictly stronger thanκ being ineffably Ramsey.
A regular cardinalκ is Ramsey if and only if[1][better source needed] for any setA ⊂κ, there is a transitive setM ⊨ ZFC− (i.e.ZFC without the axiom of powerset) of sizeκ withA ∈M, and a nonprincipalultrafilterU on the Boolean algebraP(κ) ∩ M such that:
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