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.

The existence of Ramsey cardinal implies that the existence of the zero sharp cardinal and this in turn implies the falsity of Axiom of Constructibility, of Kurt Godel.

References

• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2. 
• Erdős, Paul; Hajnal, András (1962), "Some remarks concerning our paper "On the structure of set-mappings. Non-existence of a two-valued σ-measure for the first uncountable inaccessible cardinal", Acta Mathematica Academiae Scientiarum Hungaricae 13: 223–226, doi:10.1007/BF02033641, ISSN 0001-5954, MR 0141603 
• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.