Ramsey cardinal

In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by Erdős & Hajnal (1962) and named after Frank P. Ramsey.

With [κ]<ω denoting the set of all finite subsets of κ, a cardinal number κ such that for every function

f: [κ]<ω → {0, 1}

there is a set A of cardinality κ that is homogeneous for f (i.e.: for every n, f is constant on the subsets of cardinality n from A) is called Ramsey. A cardinal κ is called almost Ramsey if for every function

f: [κ]<ω → {0, 1}

and for every λ < κ, there is a set of order type λ that is homogeneous for f.

The existence of a Ramsey cardinal is sufficient to prove the existence of 0#. In fact, if κ is Ramsey, then every set with rank less than κ has a sharp.

Every measurable cardinal is a Ramsey cardinal, and every Ramsey cardinal is a Rowbottom cardinal.

A property intermediate in strength between Ramseyness and measurability is existence of a κ-complete normal non-principal ideal I on κ such that for every A ∉ I and for every function

f: [κ]<ω → {0, 1}

there is a set B ⊂ A not in I that is homogeneous for f. If I is taken to be the ideal of nonstationary sets, this property defines the ineffably Ramsey cardinals.

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