In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by .
A cardinal <math>\kappa</math> is called <math>\alpha</math>-Erdős if for every function <math>f:[\kappa]^{<\omega} \to \{0,1\}</math>, there is a set of order type <math>\alpha</math> that is homogeneous for <math>f</math>. In the notation of the partition calculus, <math>\kappa</math> is <math>\alpha</math>-Erdős if
:<math>\kappa \rightarrow (\alpha)^{<\omega}</math>.
Under this definition, any cardinal larger than the least <math>\alpha</math>-Erdős cardinal is <math>\alpha</math>-Erdős.
The existence of zero sharp implies that the constructible universe <math>L</math> satisfies "for every countable ordinal <math>\alpha</math>, there is an <math>\alpha</math>-Erdős cardinal". In fact, for every indiscernible <math>\kappa</math>, <math>L_\kappa</math> satisfies "for every ordinal <math>\alpha</math>, there is an <math>\alpha</math>-Erdős cardinal in <math>\mathrm{Coll}(\omega, \alpha)</math>" (the Lévy collapse to make <math>\alpha</math> countable).
However, the existence of an <math>\omega_1</math>-Erdős cardinal implies existence of zero sharp. If <math>f</math> is the satisfaction relation for <math>L</math> (using ordinal parameters), then the existence of zero sharp is equivalent to there being an <math>\omega_1</math>-Erdős ordinal with respect to <math>f</math>. Thus, the existence of an <math>\omega_1</math>-Erdős cardinal implies that the axiom of constructibility is false.
The least <math>\omega</math>-Erdős cardinal is not weakly compact,<sup>p. 39.</sup> nor is the least <math>\omega_1</math>-Erdős cardinal.<sup>p. 138.</sup>
An equivalent definition is that <math>\kappa</math> is <math>\alpha</math>-Erdős if for any <math>A\subseteq\kappa</math>, there is a set <math>I</math> of order-type <math>\alpha</math> of order-indiscernibles for the structure <math>(L_\kappa[A];\in, A)</math> such that:
- for every <math>\beta\in I</math>, <math>(L_\beta[A];\in, A)\prec (L_\kappa[A];\in, A)</math>, and
- for every <math>\gamma<\kappa</math>, the set <math>I\setminus\gamma</math> forms a set of order-indiscernibles for the structure <math>(L_\kappa[A];\in, A, \xi)_{\xi<\gamma}</math>.
The least cardinal <math>\kappa</math> to satisfy the partition relation <math>\kappa \rightarrow (\alpha)^{<\omega}</math> is still <math>\alpha</math>-Erdős under this definition. Every <math>\omega</math>-Erdős cardinal is an inaccessible limit of ineffable cardinals.
See also
- List of large cardinal properties