Erdös cardinal

In mathematical logic, a cardinal number κ, which the least cardinal such that for every function f: κ ^{< ω} → {0, 1} there is a set of order type α that is homogeneous for f, is called an α-Erdős cardinal.

It is consistent with V=L that for every countable ordinal α, there is an α-Erdős cardinal. However, existence of an &omega₁-Erdős cardinal implies the existence of zero sharp.