Erdős cardinals
The $\alpha$-Erdős cardinals were introduced by Erdős and Hajnal in [1] and arose out of their study of partition relations. A cardinal $\kappa$ is $\alpha$-Erdős for an infinite limit ordinal $\alpha$ if it is the least cardinal $\kappa$ such that $\kappa\rightarrow (\alpha)^{\lt\omega}_2$ (if any such cardinal exists).
For infinite cardinals $\kappa$ and $\lambda$, the partition property $\kappa\to(\lambda)^n_\gamma$ asserts that for every function $F:[\kappa]^n\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant. Here $[X]^n$ is the set of all $n$-elements subsets of $X$. The more general partition property $\kappa\to(\lambda)^{\lt\omega}_\gamma$ asserts that for every function $F:[\kappa]^{\lt\omega}\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant for every $n$, although the value of $F$ on $[H]^n$ may be different for different $n$. Indeed, if $\kappa$ is $\alpha$-Erdős for some infinite ordinal $\alpha$, then $\kappa\rightarrow (\alpha)^{\lt\omega}_\lambda$ for all $\lambda<\kappa$ (Silver's PhD thesis). A cardinal $\kappa$ is Ramsey precisely when it is $\kappa$-Erdős.
The $\alpha$-Erdős cardinal is precisely the least cardinal $\kappa$ such that for any language $\mathcal{L}$ of size less than $\kappa$ and any structure $\mathcal{M}$ with language $\mathcal{L}$ and domain $\kappa$, there is a set of indescernibles for $\mathcal{M}$ of order-type $\alpha$.
A cardinal $\kappa$ is called Erdős if and only if it is $\alpha$-Erdős for some infinite limit ordinal $\alpha$. Because there exists at most one $\alpha$-Erdős cardinal, the notations $\eta_\alpha$ and $\kappa(\alpha)$ are sometimes used to denote the $\alpha$-Erdős cardinal.
Erdős cardinals and the constructible universe
- $\omega_1$-Erdős cardinals imply that $0^\sharp$ exists and hence there cannot be $\omega_1$-Erdős cardinals in $L$. [2]
- $\alpha$-Erdős cardinals are downward absolute to $L$ for $L$-countable $\alpha$. More generally, $\alpha$-Erdős cardinals are downward absolute to any transitive model of ZFC for $M$-countable $\alpha$. [3]
Facts
- Every Erdős cardinal is inaccessible. (Silver's PhD thesis)
- Every Erdős cardinal is subtle. [4]
- $\eta_\omega$ is a stationary limit of ineffable cardinals. [5]
- The existence of $\eta_\omega$ implies the consistency of a proper class of $n$-iterable cardinals for every $1\leq n<\omega$. [6]
- $\eta_\alpha<\eta_\beta$ whenever $\alpha<\beta$ and $\eta_\alpha\geq\alpha$. [7]
- The consistency strength of the existence of an Erdős cardinal is stronger than that of the existence of an $n$-iterable cardinal for every $n<\omega$ and weaker than that of the existence of $0^{\#}$.
- The existence of a proper class of Erdős cardinals is equivalent to the existence of a proper class of almost Ramsey cardinals. The consistency strength of this is weaker than a worldly almost Ramsey cardinal, but stronger than an almost Ramsey cardinal.
References
- Erdős, Paul and Hajnal, Andras. On the structure of set-mappings. Acta Math Acad Sci Hungar 9:111--131, 1958. MR bibtex
- Silver, Jack. Some applications of model theory in set theory. Ann Math Logic 3(1):45--110, 1971. MR bibtex
- Silver, Jack. A large cardinal in the constructible universe. Fund Math 69:93--100, 1970. MR bibtex
- Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www bibtex
- Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex
- Gitman, Victoria. Ramsey-like cardinals. The Journal of Symbolic Logic 76(2):519-540, 2011. www arχiv MR bibtex
- Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www bibtex