In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by. A cardinal \kappa is called \alpha-Erdős if for every function, there is a set of order type \alpha that is homogeneous for f. In the notation of the partition calculus, \kappa is \alpha-Erdős if The existence of zero sharp implies that the constructible universe L satisfies "for every countable ordinal \alpha, there is an \alpha-Erdős cardinal". In fact, for every indiscernible \kappa, L_\kappa satisfies "for every ordinal \alpha, there is an \alpha-Erdős cardinal in " (the Lévy collapse to make \alpha countable). However, the existence of an \omega_1-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 \omega_1-Erdős ordinal with respect to f. 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 \kappa is \alpha-Erdős, then it is \alpha-Erdős in every transitive model satisfying "\alpha is countable."

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