Erdős cardinal

In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal (1958).

A cardinal κ is called α-Erdős if for every function  f : κ^{< ω} → {0, 1}, there is a set of order type α that is homogeneous for  f . In the notation of the partition calculus, κ is α-Erdős if

κ(α) → (α)^{< ω}.

The existence of zero sharp implies that the constructible universe L satisfies "for every countable ordinal α, there is an α-Erdős cardinal". In fact, for every indiscernible κ, L_κ satisfies "for every ordinal α, there is an α-Erdős cardinal in Coll(ω, α)" (the Levy collapse to make α countable).

However, the existence of an ω₁-Erdős cardinal implies existence of zero sharp. If  f  is the satisfaction relation for L (using ordinal parameters), then the existence of zero sharp is equivalent to there being an ω₁-Erdős ordinal with respect to  f . Thus, the existence of zero sharp implies that the axiom of constructibility is false.

If κ is α-Erdős, then it is α-Erdős in every transitive model satisfying "α is countable."