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.

Let [κ]^<ω denote the set of all finite subsets of κ. Then a cardinal number κ is called Ramsey if, for every function

f: [κ]^<ω → {0, 1}

there is a set A of cardinality κ that is homogeneous for f. That is, for every n, f is constant on the subsets of cardinality n from A. A cardinal κ is called ineffably Ramsey if A can be chosen to be stationary subset of κ. A cardinal κ is called virtually Ramsey if for every function

f: [κ]^<ω → {0, 1}

there is C, a closed and unbounded subset of κ, so that for every λ in C of uncountable cofinality, there is an unbounded subset of λ, which is homogenous for f; slightly weaker is the notion of almost Ramsey where homogenous sets for f are required of order type λ, for every λ < κ.

The existence of any of these kinds of Ramsey cardinal is sufficient to prove the existence of 0^#, or indeed that 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. This is strictly stronger than κ being ineffably Ramsey.

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 Gödel.

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.