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Erdős cardinals

The $\alpha$-Erdős cardinals were introduced by Erdős and Hajnal in (Erdős & Hajnal, 1958) 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).

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.

Different terminology (Baumgartner, 1977): an infinite cardinal $κ$ is $ω$-Erdős if for every club $C$ in $κ$ and every function $f : [C]^{<ω} → κ$ that is regressive (i.e. $f(a) < \min(a)$ for all $a$ in the domain of $f$) there is a subset $X ⊂ C$ of order type $ω$ that is homogeneous for $f$ (i.e. $f ↾ [X]^n$ is constant for all $n < ω$). Schmerl, 1976 (theorem 6.1) showed that the least cardinal $κ$ such that $κ → (ω)_2^{<ω}$ has this property, if it exists.(Wilson, 2018)


With Baumgartner definition:(Wilson, 2018)

Erdős cardinals and the constructible universe:

Relations with other large cardinals:

Weakly Erdős and greatly Erdős

(Information in this section from (Sharpe & Welch, 2011))

Suppose that $κ$ has uncountable cofinality, $\mathcal{A}$ is $κ$-structure, with $X ⊆ κ$, and $t_\mathcal{A} ( X ) = \{ α ∈ κ \text{ — limit ordinal} : \text{there exists a set $I ⊆ α ∩ X$ of good indiscernibles for $\mathcal{A}$ cofinal in $α$} \}$. Using this one can define a hierarchy of normal filters $\mathcal{F}_\alpha$ potentially for all $α < κ^+$ ; these are generated by suprema of sets of nested indiscernibles for structures $\mathcal{A}$ on $κ$ using the above basic $t_\mathcal{A} (X)$ operation. A cardinal $κ$ is weakly $α$-Erdős when $\mathcal{F}_\alpha$ is non-trivial.

$κ$ is greatly Erdős iff there is a non-trivial normal filter $\mathcal{F}$ on $\mathcal{F}$ such that $F$ is closed under $t_\mathcal{A} (X)$ for every $κ$-structure $\mathcal{A}$. Equivalently (for uncountable cofinality of cardinal $κ$):

and (for inaccessible $κ$ and any choice $⟨ f_β : β < κ^+ ⟩$ of canonical functions for $κ$):



  1. Erdős, P., & Hajnal, A. (1958). On the structure of set-mappings. Acta Math. Acad. Sci. Hungar, 9, 111–131.
  2. Wilson, T. M. (2018). Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle.
  3. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag.
  4. Silver, J. (1971). Some applications of model theory in set theory. Ann. Math. Logic, 3(1), 45–110.
  5. Silver, J. (1970). A large cardinal in the constructible universe. Fund. Math., 69, 93–100.
  6. Jensen, R., & Kunen, K. (1969). Some combinatorial properties of L and V. raesch/org/jensen.html
  7. Jech, T. J. (2003). Set Theory (Third). Springer-Verlag.
  8. Gitman, V., & Shindler, R. Virtual large cardinals.
  9. Gitman, V. (2011). Ramsey-like cardinals. The Journal of Symbolic Logic, 76(2), 519–540.
  10. Sharpe, I., & Welch, P. (2011). Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Ann. Pure Appl. Logic, 162(11), 863–902.
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