# Erdős cardinals

The $\alpha$-Erdős cardinals were introduced by Erdős and Hajnal in  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.

## Facts

• $\eta_\alpha<\eta_\beta$ whenever $\alpha<\beta$ and $\eta_\alpha\geq\alpha$. 

With Baumgartner definition:

• Every $ω$-Erdős cardinal is inaccessible.
• If $η$ is an $ω$-Erdős cardinal then $η → (ω)_α^{<ω}$ for every cardinal $α < η$.
• If $α ≥ 2$ is a cardinal and there is a cardinal $η$ such that $η → (ω)_α^{<ω}$, then the least such cardinal $η$ is an $ω$-Erdős cardinal (and is greater than α.)
• Simple conclusions from the last two facts:
• The statement “there is an $ω$-Erdős cardinal” is equivalent to the statement $∃_η η → (ω)_2^{<ω}$.
• The statement “there is a proper class of $ω$-Erdős cardinals” is equivalent to the statement $∀_α ∃_η η → (ω)_α^{<ω}$.

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 $L$. 
• “$∀_{a ∈ {}^\omega\omega} \text{$a^\sharp$exists}$” is stronger than “$\text{$0^\sharp$exists}$”, but weaker then an $\omega_1$-Erdős cardinal.
• $\alpha$-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 $\alpha$. 

Relations with other large cardinals:

• Every Erdős cardinal is inaccessible. (Silver's PhD thesis)
• Every Erdős cardinal is subtle. 
• $\eta_\omega$ is a stationary limit of ineffable cardinals. 
• $\eta_\omega$ is a limit of virtually rank-into-rank cardinals. 
• If there is an $\omega$-Erdős cardinal, than there is a transitive set model of ZFC+BTEETheorem 3.5
• The existence of $\eta_\omega$ implies the consistency of a proper class of $n$-iterable cardinals for every $1\leq n<\omega$.
• For an additively indecomposable ordinal $λ ≤ ω_1$, $η_λ$ is a limit of $λ$-iterable cardinals and if there is a $λ + 1$-iterable cardinal, then there is a $λ$-Erdős cardinal below it.
• 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 $0^\sharp$.
• 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.
• The existence of an almost Ramsey cardinal is stronger than the existence of an $\omega_1$-Erdős cardinal. 
• A cardinal $\kappa$ is Ramsey precisely when it is $\kappa$-Erdős.
• If there is an $ω$-Erdős cardinal, there are $α < β < ω_1$ such that $L_β \models$ “α is remarkable”.Theorem 3.8
• (Baumgartner definition) The existence of non-remarkable weakly remarkable cardinals is equiconsistent to the existence of $ω$-Erdős cardinal (equivalent assuming $V=L$):
• Every $ω$-Erdős cardinal is a limit of non-remarkable weakly remarkable cardinals.
• If $κ$ is a non-remarkable weakly remarkable cardinal, then some ordinal greater than $κ$ is an $ω$-Erdős cardinal in $L$.

## Weakly Erdős and greatly Erdős

(Information in this section from )

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 $κ$):

• $\mathcal{G} = \bigcup_{\alpha < \kappa^+} \mathcal{F}_\alpha \not\ni \varnothing$
• $κ$ is $α$-weakly Erdős for all $α < κ^+$

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

• $\{γ < κ : f_β (γ) ⩽ o_\mathcal{A} (γ)\} \neq \varnothing$ for all $β < κ^+$ and $κ$-structures $\mathcal{A}$ such that $\mathcal{A} \models ZFC$

Relations:

• If $κ$ is a $2$-weakly Erdős cardinal then $κ$ is almost Ramsey.
• If $κ$ is virtually Ramsey then $κ$ is greatly Erdős.
• There are stationarily many completely ineffable, greatly Erdős cardinals below any Ramsey cardinal.