The $\alpha$-Erdős cardinals were introduced by Erdős and Hajnal in
{% cite Erdoes1958 %} 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%20property.md)
$\kappa\to(\lambda)^n_\gamma$ 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.{% cite Wilson2018 %}
## Facts
- $\eta_\alpha<\eta_\beta$ whenever $\alpha<\beta$ and
$\eta_\alpha\geq\alpha$.
{% cite Kanamori2009 %}
With Baumgartner
definition:{% cite Wilson2018 %}
- 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
<a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$0^\sharp
lt;/a>
exists and hence there cannot be $\omega_1$-Erdős cardinals in
$L$. {% cite Silver1971 %}
- $\alpha$-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](ZFC.md "ZFC") for
$M$-countable $\alpha$. {% cite Silver1970 %}
Relations with other large cardinals:
- Every Erdős cardinal is
[inaccessible](Inaccessible.md "Inaccessible").
(Silver's PhD thesis)
- Every Erdős cardinal is
<a href="Subtle" class="mw-redirect" title="Subtle">subtle</a>.
{% cite Jensen1969 %}
- $\eta_\omega$ is a stationary limit of
[ineffable](Ineffable.md "Ineffable")
cardinals. {% cite Jech2003 %}
- $η_ω$ is a limit of
<a href="Rank-into-rank" class="mw-redirect" title="Rank-into-rank">virtually rank-into-rank</a>
cardinals. {% cite Gitmana %}
- The existence of $\eta_\omega$ implies the consistency of a
proper class of
[$n$-iterable](Ramsey.md#iterable "Ramsey")
cardinals for every $1\leq
n<\omega$.{% cite Gitman2011 %}
- For an additively indecomposable ordinal $λ ≤ ω_1$, $η_λ$ (the
least $λ$-Erdős cardinal) 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.{% cite Gitmana %}
- 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
<a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$0^{\#}lt;/a>.
- The existence of a proper class of Erdős cardinals is equivalent to
the existence of a proper class of [](Ramsey.md#Almost_Ramsey_cardinal)
cardinals. The consistency strength of this is weaker than a
[worldly](Worldly.md "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.
{% cite Sharpe2011 %}
- A cardinal $\kappa$ is
[Ramsey](Ramsey.md "Ramsey")
precisely when it is $\kappa$-Erdős.
- (Baumgartner definition) The existence of
non-[remarkable](Remarkable.md "Remarkable")
weakly remarkable cardinals is equiconsistent to the existence of
$ω$-Erdős cardinal (equivalent assuming
$V=L$):{% cite Wilson2018 %}
- 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
{% cite Sharpe2011 %})
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](Ramsey.md "Ramsey").
- If $κ$ is virtually Ramsey then $κ$ is greatly Erdős.
- There are stationarily many completely
[ineffable](Ineffable.md "Ineffable"),
greatly Erdős cardinals below any Ramsey cardinal.