Erdős cardinal

In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by.

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 $&thinsp;f&thinsp;$. In the notation of the partition calculus, κ is α-Erdős if



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 $κ → (α)^{< ω}$" (the Levy collapse to make $α$ countable).

However, the existence of an $Coll(ω, α)$-Erdős cardinal implies existence of zero sharp. If $ω_{1}$ is the satisfaction relation for $L$ (using ordinal parameters), then the existence of zero sharp is equivalent to there being an $&thinsp;f&thinsp;$-Erdős ordinal with respect to $ω_{1}$. Thus, the existence of an $$\omega_1$$-Erdős cardinal implies that the axiom of constructibility is false.

The least $$\omega$$-Erdős cardinal is not weakly compact, p. 39. nor is the least $$\omega_1$$-Erdős cardinal. p. 39

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