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, whose theorem, called Ramsey's theorem establishes that ω enjoys a certain property that Ramsey cardinals generalize to the uncountable case.

Let [κ]<ω denote the set of all finite subsets of κ. 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, the function 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 a 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 that 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. This in turn implies the falsity of the Axiom of Constructibility of Kurt Gödel.

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.

Definition by κ-models

A regular cardinal κ is Ramsey if and only if[1][better source needed] for any set A ⊂ κ, there is a transitive set M ⊨ ZFC- (i.e. ZFC without the axiom of powerset) of size κ with A ∈ M, and a nonprincipal ultrafilter U on the Boolean algebra P(κ) ∩ M such that:

  • U is an M-ultrafilter: for any sequence ⟨Xβ : β < κ⟩ ∈ M of members of U, the diagonal intersection ΔXβ = {α < κ : ∀β < α(α ∈ Xβ)} ∈ U,
  • U is weakly amenable: for any sequence ⟨Xβ : β < κ⟩ ∈ M of subsets of κ, the set {β < κ : Xβ ∈ U} ∈ M, and
  • U is σ-complete: the intersection of any countable family of members of U is again in U.

References

  1. ^ Gitman, Victoria (2008). "Ramsey-like cardinals". arXiv:0801.4723v2 [math.LO].

Bibliography

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