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. The Erdős cardinal κ(α) is defined to be 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  (if such a cardinal exists). In the notation of the partition calculus, the Erdős cardinal κ(α) is the smallest cardinal such that κ(α) → (α)< ω If κ is α-Erdős, then it is α-Erdős in every transitive model satisfying "α is countable".