Ramsey cardinals were introduced by Erdős and Hajnal in {% cite Erdoes1962 %}.
Their consistency strength lies strictly between $0^\sharp$ and measurable cardinals.
There are many Ramsey-like cardinals with strength between weakly
compact and measurable cardinals inclusively.
{% cite Feng1990 Gitman2011 Sharpe2011 Holy2018 Nielsen2018 %}
## Ramsey cardinals
### Definitions
A cardinal $\kappa$ is Ramsey if it has the [[partition property]] $\kappa\rightarrow (\kappa)^{\lt\omega}_2$.
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 Ramsey, then $\kappa\rightarrow
(\kappa)^{\lt\omega}_\lambda$ for every $\lambda<\kappa$.
Ramsey cardinals were named in honor of Frank Ramsey, whose Ramsey
theorem for partition properties of $\omega$ motivated the
generalizations of these to uncountable cardinals. A Ramsey cardinal
$\kappa$ is exactly the
[$\kappa$-Erdős](Erdos.md "Erdos")
cardinal.
Ramsey cardinals have a number of other characterizations. They may be
characterized model theoretically through the existence of
$\kappa$-sized sets of indiscernibles for models meeting the criteria
discussed below, as well as through the existence of $\kappa$-sized
models of set theory without power set with iterable
[ultrapowers](Ultrapower.md "Ultrapower").
**Indiscernibles**: Suppose $\mathcal A=(A,\ldots)$ is a model of a
language $\mathcal L$ of size less than $\kappa$ whose universe $A$
contains $\kappa$ as a subset.
If a cardinal $\kappa$ is Ramsey, then every such model $\mathcal A$
has a $\kappa$-sized set of indiscernibles $H\subseteq\kappa$, that
is, for every formula $\varphi(\overline x)$ of $\mathcal L$ and
every pair of tuples $\overline \alpha$ and $\overline \beta$ of
elements of $H$, we have $\mathcal A\models\varphi (\overline
\alpha)\leftrightarrow \varphi(\overline \beta)$.
{% cite Jech2003 %}
**Good sets of indiscernibles**: Suppose $A\subseteq\kappa$ and
$L_\kappa\[A\]$ denotes the $\kappa^{\text{th}}$-level of the
universe constructible using a predicate for $A$. A set
$I\subseteq\kappa$ is a good set of indiscernibles for the model
$\langle L_\kappa\[A\],A\rangle$ if for all $\gamma\in I$,
- $\langle L_\gamma\[A\cap \gamma\],A\cap \gamma\rangle\prec
\langle L_\kappa\[A\], A\rangle$,
- $I\setminus\gamma$ is a set of indiscernibles for the model
$\langle L_\kappa\[A\], A,\xi\rangle_{\xi\in\gamma}$.
A cardinal $\kappa$ is Ramsey if and only if for every
$A\subseteq\kappa$, there is a $\kappa$-sized good set of
indiscernibles for the model $\langle L_\kappa\[A\], A\rangle$.
{% cite Dodd1981 %}
**$M$-<a href="Ultrafilter" class="mw-redirect" title="Ultrafilter">ultrafilters</a>**:
Suppose a transitive $M\models {\rm ZFC}^-$, the theory ${\rm ZFC}$
without the power set axiom (and using collection and separation rather
than merely replacement) and $\kappa$ is a cardinal in $M$. We call
$U\subseteq P(\kappa)^M$ an $M$-ultrafilter if the model $\langle
M,U\rangle\models$“$U$ is a normal ultrafilter on $\kappa$”. In the
case when the $M$-ultrafilter is not an element of $M$, the model
$\langle M,U\rangle$ of $M$ together with a predicate for $U$ often
fails to satisfy much of ${\rm ZFC}$. An $M$-ultrafilter $U$ is said to
be weakly amenable (to $M$) if for every $A\in M$ of size $\kappa$ in
$M$, the intersection $A\cap U$ is an element of $M$. An
$M$-ultrafilter $U$ is countably complete if every countable sequence
(possibly external to $M$) of elements of $U$ has a non-empty
intersection (even if the intersection is not itself an element of $M$).
A weak $\kappa$-model is a transitive set $M\models {\rm ZFC}^- $ of
size $\kappa$ and containing $\kappa$ as an element. A modified
ultrapower construction using only functions on $\kappa$ that are
elements of $M$ can be carried out with an $M$-ultrafilter. If the
$M$-ultrafilter happens to be countably complete, then the standard
argument shows that the ultrapower is well-founded. If the
$M$-ultrafilter is moreover weakly amenable, then a weakly amenable
ultrafilter on the image of $\kappa$ in the well-founded ultrapower can
be constructed from images of the pieces of $U$ that are in $M$. The
ultrapower construction may be iterated in this manner, taking direct
limits at limit stages, and in this case the countable completeness of
the $M$-ultrafilter ensures that every stage of the iteration produces a
well-founded model. {% cite Kanamori2009 %} (Ch.
19)
A cardinal $\kappa$ is Ramsey if and only if every $A\subseteq\kappa$
is contained in a weak $\kappa$-model $M$ for which there exists a
weakly amenable countably complete $M$-ultrafilter on $\kappa$.
{% cite Dodd1981 %}
### Ramsey cardinals and the constructible universe
Ramsey cardinals imply that
<a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$0^\sharp
lt;/a>
exists and hence there cannot be Ramsey cardinals in $L$.
{% cite Kanamori2009 %}
### Relations with other large cardinals
- [Measurable](Measurable.md "Measurable")
cardinals are Ramsey and stationary limits of Ramsey cardinals.
{% cite Erdoes1962 %}
- Ramsey cardinals are
[unfoldable](Unfoldable.md "Unfoldable")
(using the $M$-ultrafilters characterization) and stationary limits
of unfoldable cardinals (as they are stationary limits of
$\omega_1$-iterable cardinals).
- Ramsey cardinals are stationary limits of
<a href="Completely_ineffable" class="mw-redirect" title="Completely ineffable">completely ineffable</a>
cardinals, they are
<a href="Weakly_ineffable" class="mw-redirect" title="Weakly ineffable">weakly ineffable</a>,
but the least Ramsey cardinal is not ineffable. Ineffable Ramsey
cardinals are limits of Ramsey cardinals, because ineffable
cardinals are $Π^1_2$-indescribable and being Ramsey is a
$Π^1_2$-statement.{% cite Gitman2011 %}
- There are stationarily many completely ineffable, [](Erdos.md)
cardinals below any Ramsey
cardinal.{% cite Sharpe2011 %}
Weaker Ramsey-like:
- The existence of a Ramsey cardinal is stronger than the existence of
a proper class of almost Ramsey cardinals.
- The Ramsey cardinals are precisely the
[Erdős](Erdos.md "Erdos")
almost Ramsey cardinals and also precisely the [[weakly compact]]
almost Ramsey cardinals.
- A Ramsey cardinal is $\omega_1$-iterable and a stationary limit of
$\omega_1$-iterable cardinals. This is already true of an
$\omega_1$-[Erdős](Erdos.md "Erdos")
cardinal. {% cite Sharpe2011 %}
- A virtually Ramsey cardinal that is [[weakly compact]]
is already Ramsey. If $κ$ is Ramsey, then there is a forcing
extension destroying this, while preserving that $κ$ is virtually
Ramsey. It is open whether virtually Ramsey cardinals are weaker
than Ramsey
cardinals. {% cite Gitman2011a Gitman %}
Stronger Ramsey-like:
- If $κ$ is $Π_1$-Ramsey, then the set of Ramsey cardinals less then
$κ$ is in the $Π_1$-Ramsey filter on
$κ$.{% cite Feng1990 %}
- Strongly Ramsey cardinals are Ramsey and stationary limits of Ramsey
cardinals.{% cite Gitman2011 %}
- Mahlo–Ramsey cardinals are a direct strengthening of
Ramseyness.{% cite Sharpe2011 %}
### Ramsey cardinals and forcing
- Ramsey cardinals are preserved by small forcing.
{% cite Kanamori2009 %}
- Ramsey cardinals $\kappa$ are preserved by the canonical forcing of
the ${\rm GCH}$, by fast function forcing, and by the forcing to
add a slim $\kappa$-Kurepa tree.
{% cite Gitman %}
- If $\kappa$ is Ramsey, there is a forcing extension in which
$\kappa$ remains Ramsey and
$2^\kappa\gt\kappa$. {% cite Gitman Cody2015 %}
- If the ${\rm GCH}$ holds and $F$ is a class function on the
regular cardinals having a closure point at $\kappa$ and
satisfying $F(\alpha)\leq F(\beta)$ for $\alpha<\beta$
and $\text{cf}(F(\alpha))>\alpha$, then there is a
cofinality preserving forcing extension in which $\kappa$
remains Ramsey and $2^\delta=F(\delta)$ for every regular
cardinal $\delta$.
{% cite Cody2015 %}
- There is a forcing extension in which $κ$ is the first cardinal
at which the $\mathrm{GCH}$ fails.
{% cite Gitman %}
- If the existence of Ramsey cardinals is consistent with ZFC, then
there is a model of ZFC in which $\kappa$ is not Ramsey, but
becomes Ramsey in a forcing extension.
{% cite Gitman %}
## Completely Romsey cardinals etc.
(All information in this section are from
{% cite Feng1990 %} unless otherwise noted)
### Basic definitions
- $\mathcal{P}(x)$ is the powerset (set of all subsets) of $x$.
$\mathcal{P}_k(x)$ is the set of all subsets of $x$ with exactly
$k$ elements.
- $f:\mathcal{P}_k(λ) \to λ$ is regressive iff for all $A \in
\mathcal{P}_k(λ)$, we have $f(A) < \min(A)$.
- $E$ is $f$-homogenous iff $E \subseteq λ$ and for all $B,C \in
\mathcal{P}_k(E)$, we have $f(B) = f(C)$.
### $Π_α$-Ramsey and completely Ramsey
Suppose that $κ$ is a regular uncountable cardinal and $I \supseteq
\mathcal{P}_{<κ}(κ)$ is an ideal on $κ$. For every $X \subseteq
$κ, $X \in \mathcal{R}^+(I)$ iff for every regressive function
$f:\mathcal{P}_{<ω}(κ) \to κ$, for every club $C \subseteq κ$,
there is a $Y \in I^+f$ such that $Y \subseteq X \cap C$ and $Y$ is
homogeneous for $f$.
$\mathcal{R}(I) = \mathcal{P}(κ) - \mathcal{R}^+(I)$
$\mathcal{R}^*(I) = \{ X \subseteq κ : κ - X \in \mathcal{R}(I)
\}$
A regular uncountable cardinal $κ$ is Ramsey iff $κ \not\in
\mathcal{R}(\mathcal{P}_{<κ}(κ))$. If it is Ramsey, we call
$\mathcal{R}(\mathcal{P}_{<κ}(κ))$ *the Ramsey
<a href="Ideal" class="mw-redirect" title="Ideal">ideal</a>*
on $κ$, its dual $\mathcal{R}^*(\mathcal{P}_{<κ}(κ))$ *the Ramsey
[filter](Filter.md "Filter")*
and every element of $\mathcal{R}^+(\mathcal{P}_{<κ}(κ))$ *a
Ramsey subset* of $κ$.
For a regular uncountable cardinal $κ$, we define
- $I_{-2}^κ = \mathcal{P}_{<κ}(κ)$
- $I_{-1}^κ = NS_κ$ (the set of non-stationary subsets of $κ$)
- for $n < ω$, $I_n^κ = \mathcal{R}(I_{n-2}^κ)$
- for $α \geq ω$, $I_{α+1}^κ = \mathcal{R}(I_α^κ)$
- for limit ordinal $γ$, $I_γ^κ = \bigcup_{β<γ}
\mathcal{R}(I_β^κ)$
Regular uncountable cardinal $κ$ is **$Π_α$-Ramsey** iff $κ \not\in
I_α^κ$ and **completely Ramsey** iff for all $α$, $κ \not\in I_α^κ$.
If $κ$ is $Π_α$-Ramsey, we call $I_α^κ$ *the $Π_α$-Ramsey ideal* on
$κ$, its dual *the $Π_α$-Ramsey filter* and every subset of $κ$ not in
$I_α^κ$ *a $Π_α$-Ramsey subset*. If $κ$ is completely Ramsey, we call
$I_{θ_κ}^κ$ *the completely Ramsey ideal* on $κ$, its dual *the
completely Ramsey filter* and every subset of $κ$ not in $I_{θ_κ}^κ$
*a completely Ramsey subset*. ($θ_κ$ is the least $α$ such that $I_α^κ
= I_{α+1}^κ$ — it must exist before $(2^κ)^+$ <span class="small">for
every regular uncountable $κ$ — even if the ideals are trivial</span>)
### $α$-hyper completely Ramsey and super completely Ramsey
A sequence $⟨f_α:α<κ^+⟩$ of elements of $^κκ$ is a *canonical
sequence* on $κ$ if both
- for all $α, β\in κ$, $α < β$ implies $f_α < f_β$.
- and for any other sequence $⟨g_α:α<κ^+⟩$ of elements of $κ^κ$
such that $\forall_{α < β < κ^κ} g_α < g_β$, we have
$\forall_{α < κ^+} f_α < g_α$.
Note four facts:
- If $⟨f_α:α<κ^+⟩$ and $⟨g_α:α<κ^+⟩$ both are canonical
sequences on $κ$, then for all $α < κ^+$ there is a club $C_α
\subseteq κ$ such that $\forall_{γ \in C_α} f_α(γ) = g_α(γ)$.
(*All pairs of corresponding elements of two sequences of functions
are equal on a club.*)
- There are canonical sequences on each regular uncountable cardinal.
- If $⟨h_α:α<κ^+⟩$ is a canonical sequence on $κ$, then for all $α
< κ^+$ there is a club $C_α \subseteq κ$ such that
$\forall_{η \in C_α} h_α(η) < \|η\|^+$. (*Each function in a
sequence takes on a club values with cardinality not greater then
argument's.*)
- For all $β < κ^+$ there is a club $C_β \subseteq κ$ such that
for all uncountable regular $λ \in C_β$, the set $\{ γ < λ :
f^λ_{f^κ_β(λ)}(γ) = f^κ_β(γ) \}$ contains a club in $λ$, where
$\vec {f^λ}$ and $\vec {f^κ}$ are canonical sequences on $λ$ and
$κ$ respectively.
For a regular uncountable cardinal $κ$, let $\vec f = ⟨f_α:α<κ^+⟩$
be the canonical sequence on $κ$.
- $κ$ is **0-hyper completely Ramsey** iff $κ$ is completely Ramsey.
- For $α<κ^+$, $κ$ is **$α+1$-hyper completely Ramsey** iff $κ$ is
$α$-hyper completely Ramsey and there is a completely Ramsey subset
$X$ such that for all $λ \in X$, $λ$ is $f_α(λ)$-hyper completely
Ramsey.
- For $γ \leq κ^+$, $κ$ is **$γ$-hyper completely Ramsey** iff $κ$ is
$β$-hyper completely Ramsey for all $β<γ$.
- $κ$ is **super completely Ramsey** iff $κ$ is $κ^+$-hyper completely
Ramsey.
### Terminology
(This subsection compares (Sharpe&Welch, 2011) and (Feng, 1990))
$Π_α$-Ramsey cardinals correspond to $α$-Ramsey and $α$-Ramsey$^s$ in
{% cite Sharpe2011 Holy2018 %}
(The “$^s$” stands for
“stationary”.{% cite Sharpe2011 %})
$Π_{2 n}$-Ramsey cardinals are Sharpe-Welch $n$-Ramsey and $Π_{2 n +
1}$-Ramsey cardinals are $n$-Ramsey$^s$.
For infinite $α$, $Π_α$-Ramsey, Sharpe-Welch $α$-Ramsey and
$α$-Ramsey$^s$ are identical.
### Results
Absoluteness:
- All this properties (being Ramsey itself, $Π_α$-Ramsey, completely
Ramsey, $α$-hyper completely Ramsey and super completely Ramsey) are
downwards absolute to the Dodd-Jensen [](Core%20models.md).
Hierarchy:
- There are stationary many $Π_n$-Ramsey cardinals below each
$Π_{n+1}$-Ramsey cardinal.
- If $κ$ is $Π_{α+1}$-Ramsey and $α < κ^+$, then the set of
$Π_α$-Ramsey cardinals less then $κ$ is in the $Π_{α+1}$-Ramsey
filter on $κ$.
Upper limit of consistency strength:
- Any
[measurable](Measurable.md "Measurable")
cardinal is super completely Ramsey and a stationary limit of super
completely Ramsey cardinals.
Indescribability:
- If $κ$ is $Π_n$-Ramsey, then $κ$ is
$Π_{n+1}^1$-[indescribable](Indescribable.md "Indescribable").
If $X \subseteq κ$ is a $Π_n$-Ramsey subset, then $X$ is
$Π_{n+1}^1$-indescribable.
- For infinite $α$, if $κ$ is $Π_α$-Ramsey, then $κ$ is $Π^1_{2
·(1+β)+ 1}$-indescribable for each $β < \min \{α, κ^+\}$
(Transfinite $Π^1_α$-indescribable is defined via finite
games.).{% cite Sharpe2011 %}
- If $κ$ is completely Ramsey, then $κ$ is
$Π_1^2$-[indescribable](Indescribable.md "Indescribable").{% cite Holy2018 %}
Equivalence:
- A cardinal is completely Ramsey iff it is $ω$-very
Ramsey. {% cite Sharpe2011 Nielsen2018 %}
Relation with other variants of Ramseyness:
- Strongly Ramsey cardinals are limits of completely Ramsey cardinals,
but are not necessarily completely Ramsey
themselves.{% cite Gitman2011a %}
- Every $(ω+1)$-Ramsey cardinal is a completely Ramsey stationary
limit of completely Ramsey
cardinals.{% cite Nielsen2018 %}
- Any $\Pi_2$-Ramsey cardinal is $α$-Mahlo–Ramsey for all $α <
κ^+$. {% cite Sharpe2011 %}
## Almost Ramsey cardinal
cf. (Vickers&Welch, 2001)
An uncountable cardinal $\kappa$ is **almost Ramsey** if and only if
$\kappa\rightarrow(\alpha)^{<\omega}$ for every
$\alpha<\kappa$. Equivalently:
- $\kappa\rightarrow(\alpha)^{<\omega}_\lambda$ for every
$\alpha,\lambda<\kappa$
- For every structure $\mathcal{M}$ with language of size
lt;\kappa$, there is are sets of indiscernibles
$I\subseteq\kappa$ for $\mathcal{M}$ of any size lt;\kappa$.
- For every $\alpha<\kappa$, $\eta_\alpha$ exists and
$\eta_\alpha<\kappa$.
- $\kappa=\text{sup}\{\eta_\alpha:\alpha<\kappa\}$
($\eta_\alpha$ is the
[$\alpha$-Erdős](Erdos.md "Erdos")
cardinal.)
Every almost Ramsey cardinal is a [](Beth.md), but
the least almost Ramsey cardinal, if it exists, has cofinality
$\omega$. In fact, the least almost Ramsey cardinal is not
<a href="Weakly_inaccessible" class="mw-redirect" title="Weakly inaccessible">weakly inaccessible</a>,
[worldly](Worldly.md "Worldly"),
or
<a href="Correct" class="mw-redirect" title="Correct">correct</a>.
However, if the least almost Ramsey cardinal exists, it is larger than
the least
[$\omega_1$-Erdős](Erdos.md "Erdos")
cardinal. Any regular almost Ramsey cardinal is worldly, and any worldly
almost Ramsey cardinal $\kappa$ has $\kappa$ almost Ramsey cardinals
below it.
The existence of a worldly almost Ramsey cardinal is stronger than the
existence of a proper class of almost Ramsey cardinals. Therefore, the
existence of a Ramsey cardinal is stronger than the existence of a
proper class of almost Ramsey cardinals. The existence of a proper class
of almost Ramsey cardinals is equivalent to the existence of
$\eta_\alpha$ for every $\alpha$. The existence of an almost Ramsey
cardinal is stronger than the existence of an $\omega_1$-Erdős
cardinal.
The existence of an almost Ramsey cardinal is equivalent to the
existence of $\eta^n(\omega)$ for every $n<\omega$. On one hand,
if a almost Ramsey cardinal $\kappa$ exists, then $\omega<\kappa$.
Then, $\eta_\omega$ is less than $\kappa$. Then,
$\eta_{\eta_\omega}$ exists and is less than $\kappa$, and so on.
On the other hand, if $\eta^n(\omega)$ exists for every
$n<\omega$, then $\text{sup}\{\eta^n(\omega):n<\omega\}$ is
almost Ramsey, and in fact the least almost Ramsey cardinal. Note that
such a set exists by replacement and has a supremum by union.
The Ramsey cardinals are precisely the
[Erdős](Erdos.md "Erdos")
almost Ramsey cardinals and also precisely the [[weakly compact]]
almost Ramsey cardinals.
If $κ$ is a $2$-weakly Erdős cardinal, then $κ$ is almost
Ramsey.{% cite Sharpe2011 %}
## Strongly Ramsey cardinal
Strongly Ramsey cardinals were introduced by Gitman in
{% cite Gitman2011 %} (all information from
there unless otherwise noted). They strengthen the $M$-ultrafilters
characterization of Ramsey cardinals from weak $\kappa$-models to
$\kappa$-models.
A cardinal $\kappa$ is **strongly Ramsey** if every
$A\subseteq\kappa$ is contained in a $\kappa$-model $M$ for which
there exists a weakly amenable $M$-ultrafilter on $\kappa$. An
$M$-ultrafilter for a $\kappa$-model $M$ is automatically countably
complete since $\langle M,U\rangle$ satisfies that it is
$\kappa$-complete and it must be correct about this since $M$ is closed
under sequences of length less than $\kappa$.
Properties:
- Super Ramsey cardinals are strongly Ramsey limits of strongly Ramsey
cardinals.
- Strongly Ramsey cardinals are limits of completely Ramsey cardinals,
but are not necessarily completely Ramsey
themselves.{% cite Gitman2011a %}
- Every strongly Ramsey cardinal is a stationary limit of almost fully
Ramseys.{% cite Nielsen2018 %}
- Strongly Ramsey cardinals are Ramsey and stationary limits of Ramsey
cardinals.
- The least strongly Ramsey cardinal is not
[ineffable](Ineffable.md "Ineffable").
- Forcing related properties of strongly Ramsey cardinals are the same
as those of Ramsey cardinals described above.
{% cite Gitman %}
- Strong Ramseyness is downward absolute to $K$.
{% cite Gitman2011a %}
## Super Ramsey cardinal
Super Ramsey cardinals were introduced by Gitman in
{% cite Gitman2011 %} (all information from
there unless otherwise noted). They strengthen one definition of strong
Ramseyness.
A weak $\kappa$-model $M$ is a $\kappa$-model if additionally
$M^{\lt\kappa}\subseteq M$.
A cardinal $\kappa$ is **super Ramsey** if and only if for every
$A\subseteq\kappa$, there is some $\kappa$-model $M$ with
$A\subseteq M\prec H_{\kappa^+}$ such that there is some $N$ and
some $\kappa$-powerset preserving nontrivial elementary embedding
$j:M\prec N$.
The following are some facts about super Ramsey cardinals:
- [Measurable](Measurable.md "Measurable")
cardinals are super Ramsey limits of super Ramsey cardinals.
- Fully Ramsey cardinals are super Ramsey limits of super Ramsey
cardinals.{% cite Holy2018 %}
- Super Ramsey cardinals are strongly Ramsey limits of strongly Ramsey
cardinals.
- Super Ramseyness is downward absolute to $K$.
{% cite Gitman2011a %}
- The required $M$ for a super Ramsey embedding is stationarily
correct.
## $\alpha$-iterable cardinal
The $\alpha$-iterable cardinals for $1\leq\alpha\leq\omega_1$ were
introduced by Gitman in
{% cite Gitman2011a %}. They form a
hierarchy of large cardinal notions strengthening [[weakly compact]]
cardinals, while weakening the $M$-ultrafilter characterization of
Ramsey cardinals. Recall that if $\kappa$ is Ramsey, then every
$A\subseteq\kappa$ is contained in a weak $\kappa$-model $M$ for
which there exists an $M$-ultrafilter, the ultrapower construction with
which may be iterated through all the ordinals. Suppose $M$ is a weak
$\kappa$-model and $U$ is an $M$-ultrafilter on $\kappa$. Define that:
- $U$ is $0$-good if the ultrapower is well-founded,
- $U$ is 1-good if it is 0-good and weakly amenable,
- for an ordinal $\alpha>1$, $U$ is $\alpha$-good, if it produces
at least $\alpha$-many well-founded iterated ultrapowers.
Using a theorem of Gaifman
{% cite Gaifman1974 %}, if an
$M$-ultrafilter is $\omega_1$-good, then it is already $\alpha$-good
for every ordinal $\alpha$.
For $1\leq\alpha\leq\omega_1$, a cardinal $\kappa$ is
**$\alpha$-iterable** if every $A\subseteq\kappa$ is contained in a
weak $\kappa$-model $M$ for which there exists an $\alpha$-good
$M$-ultrafilter on $\kappa$.
The $\alpha$-iterable cardinals form a hierarchy of strength above
weakly compact cardinals and below Ramsey cardinals.
The $1$-iterable cardinals are sometimes called the **weakly Ramsey**
cardinals.
### Results
Lower limit:
- $1$-iterable cardinals are
<a href="Weakly_ineffable" class="mw-redirect" title="Weakly ineffable">weakly ineffable</a>
and stationary limits of
<a href="Completely_ineffable" class="mw-redirect" title="Completely ineffable">completely ineffable</a>
cardinals. The least $1$-iterable cardinal is not ineffable.
{% cite Gitman2011 %}
- Super weakly Ramsey cardinals are weakly Ramsey (=$1$-iterable)
limits of weakly Ramsey cardinals.
Upper limit:
- A Ramsey cardinal is $\omega_1$-iterable and a stationary limit of
$\omega_1$-iterable cardinals. This is already true of an
$\omega_1$-[Erdős](Erdos.md "Erdos")
cardinal. {% cite Sharpe2011 %}
- If $C ∈ V\[H\]$, a forcing extension by $\mathrm{Coll}(ω, V_κ)$,
is a club in $κ$ of generating indiscernibles for $V_κ$ of
order-type $κ$ (like in the definition of
<a href="Silver_cardinal" class="mw-redirect" title="Silver cardinal">Silver cardinals</a>),
then each $ξ ∈ C$ is lt;
ω_1$-iterable.{% cite Gitmana %}
- $ω_1$-iterable cardinals are
<a href="Strongly_unfoldable" class="mw-redirect" title="Strongly unfoldable">strongly unfoldable</a>
in $L$.{% cite Gitman2011a %}
Hierarchy:
- An $\alpha$-iterable cardinal is $\beta$-iterable and a stationary
limit of $\beta$-iterable cardinals for every $\beta<\alpha$.
{% cite Gitman2011a %}
- For $β > 0$, every $(α, β)$-Ramsey is a $β$-iterable stationary
limit of $β$-iterables.
- It is consistent from an
$\omega$-[Erdős](Erdos.md "Erdos")
cardinal that for every $n\in\omega$, there is a proper class of
$n$-iterable cardinals.
- 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 %}
- A [](Huge.md)
cardinal is an $n+1$-iterable limit of $n+1$-iterable cardinals. If
$κ$ is $n+2$-iterable, then $V_κ$ is a model of proper class many
virtually $n$-huge*
cardinals.{% cite Gitmana %}
- Every virtually rank-into-rank cardinal is an
$ω$-<a href="Iterable" class="mw-redirect" title="Iterable">iterable</a>
limit of $ω$-iterable
cardinals.{% cite Gitmana %}
Between $1$- and $2$-iterable:
- A $2$-iterable cardinal implies the consistency of a
[remarkable](Remarkable.md "Remarkable")
cardinal: Every $2$-iterable cardinal is a limit of remarkable
cardinals. {% cite Gitman2011a %}
- Remarkable cardinals imply the consistency of $1$-iterable
cardinals: If there is a remarkable cardinal, then there is a
countable transitive model of ZFC with a proper class of
$1$-iterable cardinals.
{% cite Gitman2011a %}
- If $κ$ is $2$-iterable, then $V_κ$ is a model of proper class many
[](Extendible.md)
cardinals for every $n < ω$, of proper class many
<a href="Shelah" class="mw-redirect" title="Shelah">virtually Shelah for supercompactness</a>
cardinals{% cite Gitmana %} and
of proper class many
<a href="Completely_remarkable" class="mw-redirect" title="Completely remarkable">completely remarkable</a>
cardinals.{% cite Bagaria2017a %}
- Virtually extendible cardinals are 1-iterable limits of 1-iterable
cardinals.{% cite Gitmana %}
Absoluteness:
- $\omega_1$-iterable cardinals imply that
<a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$0^\sharplt;/a>
exists and hence there cannot be $\omega_1$-iterable cardinals in
$L$. For $L$-countable $\alpha$, the $\alpha$-iterable cardinals
are downward absolute to $L$. In fact, if
<a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$0^\sharplt;/a>
exists, then every Silver indiscernible is $\alpha$-iterable in $L$
for every $L$-countable $\alpha$.
{% cite Gitman2011a %}
- $\alpha$-iterable cardinals $\kappa$ are preserved by small
forcing, by the canonical forcing of the ${\rm GCH}$, by fast
function forcing, and by the forcing to add a slim $\kappa$-Kurepa
tree. If $\kappa$ is $\alpha$-iterable, there is a forcing
extension in which $\kappa$ remains $\alpha$-iterable and
$2^\kappa\gt\kappa$.
{% cite Gitman %}
## Mahlo–Ramsey cardinals
The property of being Mahlo–Ramsey (MR) is a slight strengthening of
Ramseyness introduced in analogy to
[Mahlo](Mahlo.md "Mahlo")
cardinals in {% cite Sharpe2011 %} (all
information from there).
For a regular cardinal $κ$ and a sequence of canonical functions $⟨ f_α
: α < κ^+ ⟩$
- $κ$ is $0$-MR iff it is Ramsey.
- $κ$ is $(α + 1 )$-MR iff for any $g : \mathcal{P}_{<ω}(κ) → 2$
there is an $X ∈ NS^+_κ$ such that $X$ is homogeneous for $g$ and
$∀_{μ ∈ X} \text{$μ$ is $f_α (μ)$-MR}$.
- $κ$ is $δ$-MR for limit $δ < κ^+$ iff it is $α$-MR for all $α
< δ$.
Any $\Pi_2$-Ramsey cardinal is $α$-MR for all $α < κ^+$.
## Very Ramsey cardinals
For $X ⊆ κ$ and ordinal $α$, $G_R(X, α)$ is a certain game for two
players with finitely many moves defined in (Sharpe&Welch11). $X$ is
Sharpe-Welch $\alpha$-Ramsey iff (II) wins $G_R(X, α)$. $G_r(X, α)$
(also defined there) is a modification of the game allowing $1+α$ moves.
$X$ is **$\alpha$-very Ramsey** iff (II) has a winning strategy in
$G_r(X, α)$.{% cite Sharpe2011 %}
For $n < ω$, the games $G_R(X, n)$ and $G_r(X, n)$
coincide.{% cite Sharpe2011 %}
In analogy to coherent lt;α$-very Ramsey, one can define coherent
lt;α$-very Ramsey cardinals. $α$-very Ramsey cardinals are equivalent
to coherent lt;α$-very Ramsey cardinals for limit $α$ and to
lt;(α+1)$-very Ramsey cardinals in general. (This just allows to
“subtract one” for successor
ordinals.){% cite Nielsen2018 %}
Results:
- A cardinal is completely Ramsey iff it is $ω$-very
Ramsey. {% cite Sharpe2011 Nielsen2018 %}
- If $κ$ is a
[measurable](Measurable.md "Measurable")
cardinal, then $κ$ is $κ$-very
Ramsey.{% cite Sharpe2011 %}
- If a cardinal is $ω_1$-very Ramsey (=strategic $ω_1$-Ramsey
cardinal), then it is measurable in the [](Core%20models.md)
unless
<a href="Zero_pistol" class="mw-redirect" title="Zero pistol">$0^\Plt;/a>
exists and an inner model with a
[Woodin](Woodin.md "Woodin")
cardinal exists. {% cite Sharpe2011 Nielsen2018 %}
Additional results from
{% cite Nielsen2018 %}:
- For limit ordinal $α$, every coherent lt;ωα$-Ramsey is $ωα$-very
Ramsey.
- For any ordinal $α$, every coherent lt;α$-very Ramsey is coherent
lt;α$-Ramsey.
- For limit ordinal $α$, $κ$ is $ωα$-very Ramsey iff it is coherent
lt;ωα$-Ramsey.
- $κ$ is $λ$-very Ramsey iff it is strategic $λ$-Ramsey for any $λ$
with uncountable cofinality.
## Virtually Ramsey cardinal
Virtually Ramsey cardinals were introduced by Sharpe and Welch in
{% cite Sharpe2011 %}. They weaken the good
indiscernibles characterization of Ramsey cardinals and were motivated
by finding an upper bound on the consistency strength of a variant of
Chang's Conjecture studied in
{% cite Sharpe2011 %}. For
$A\subseteq\kappa$, define that $\mathscr
I=\{\alpha<\kappa\mid$ there is an unbounded good set of
indiscernibles $I_\alpha\subseteq\alpha$ for $\langle
L_\kappa\[A\],A\rangle\}$. A cardinal $\kappa$ is virtually Ramsey
if for every $A\subseteq\kappa$, the set $\mathscr I$ contains a club
of $\kappa$.
Virtually Ramsey cardinals are
[Mahlo](Mahlo.md "Mahlo") and
a virtually Ramsey cardinal that is [[weakly compact]]
is already Ramsey. If $κ$ is Ramsey, then there is a forcing extension
destroying this, while preserving that $κ$ is virtually Ramsey. It is
open whether virtually Ramsey cardinals are weaker than Ramsey
cardinals. {% cite Gitman2011a Gitman %}
If κ is virtually Ramsey then κ is [](Erdos.md).{% cite Sharpe2011 %}
## Super weakly Ramsey cardinal
(All from {% cite Holy2018 %})
A cardinal $κ$ is **super weakly Ramsey** iff every $A ⊆ κ$ is
contained, as an element, in a weak
κ-[model](Model.md "Model") $M
≺ H(κ^+)$ for which there exists a $κ$-powerset preserving elementary
embedding $j∶ M → N$.
Strength:
- Super weakly Ramsey cardinals are weakly Ramsey (=$1$-iterable)
limits of weakly Ramsey cardinals.
- Super weakly Ramsey cardinals are
[ineffable](Ineffable.md "Ineffable").
- $ω$-Ramsey cardinals are super weakly Ramsey limits of super weakly
Ramsey cardinals.
## $α$-Ramsey cardinal etc.
### $α$-Ramsey cardinal for cardinal $α$
(All from {% cite Holy2018 %})
For regular cardinal $α ≤ κ$, $κ$ is $α$-Ramsey iff for arbitrarily
large regular cardinals $θ$, every $A ⊆ κ$ is contained, as an element,
in some weak $κ$-model $M ≺ H(θ)$ which is closed under
lt;α$-sequences, and for which there exists a $κ$-powerset preserving
elementary embedding $j∶ M → N$.
Note that, in the case $α = κ$, a weak $κ$-model closed under
lt;κ$-sequences is exactly a $κ$-model.
Alternate characterisation:
- For regular uncountable cardinal $α ≤ κ$, $κ$ is $α$-Ramsey iff $κ =
κ^{<κ}$ has the $α$-filter property.
- $κ$ is $ω$-Ramsey iff $κ = κ^{<κ}$ has the well-founded
$ω$-filter property.
This characterisation works better for singular alpha $α$ (the original
one would imply being $α^+$-Ramsey; well-founded $α$-filter property is
better for countable cofinality).
### Games for definitions
(from {% cite Nielsen2018 %} unless otherwise
noted)
For a weak
$κ$-[model](Model.md "Model")
$\mathcal{M}$, $μ$ is an
*$\mathcal{M}$-<a href="Measure" class="mw-redirect" title="Measure">measure</a>*
iff $(\mathcal{M}, \in, μ) \models \text{“$μ$ is a $κ$-complete
ultrafilter on $κ$”}$.
Games $G_1$ and $G_2$ are *equivalent* when each of two players has a
winning strategy in $G_1$ if and only if he has one in $G_2$.
The $α$-Ramsey cardinals are based upon *well-founded filter
games*{% cite Holy2018 %}
$wfG^θ_γ(κ)$ (full definition in sources).
- Player I
(*challenger*{% cite Holy2018 %})
gives $\subseteq$-increasing $κ$-models $\mathcal{M}_α ≺ H_θ$,
- player II
(*judge*{% cite Holy2018 %}) gives
$\subseteq$-increasing filters $μ_α$ that are
$\mathcal{M}_α$-measures
- and II wins iff after $γ$ rounds $μ$ is an $\mathcal{M}$-normal
good $\mathcal{M}$-measure for $μ = \bigcup_{α<γ} μ_α$ and
$\mathcal{M} = \bigcup_{α<γ} \mathcal{M}_α$.
The games $wfG^{θ_0}_γ(κ)$ and $wfG^{θ_1}_γ(κ)$ are equivalent for
any $γ$ with $\mathrm{cof}(γ) \neq ω$ and any regular $θ_0, θ_1 <
κ$.
$\mathcal{G}^θ_γ(κ, ζ)$ is a similar family of games (again full
definition in sources).
- Each of them lasts up to $γ+1$ rounds
- and player II wins when he does not have to end the game before
$γ+1$ rounds pass
- (I gives $\subseteq$-increasing weak $κ$-models
- and II must give normal $\mathcal{M}_α$-measures with additional
requirements for limit $α$ (eg. $μ_α$ is $ζ$-good) and for the last
move).
For convenience
- $\mathcal{G}^θ_γ(κ) := \mathcal{G}^θ_γ(κ, 0)$
- $\mathcal{G}_γ(κ) := \mathcal{G}^θ_γ(κ)$ whenever
$\mathrm{cof}(γ) \neq ω$ as again the existence of winning
strategies in these games does not depend upon a specific $θ$.
$\mathcal{G}^θ_γ(κ)$, $\mathcal{G}^θ_γ(κ, 1)$ and $wfG^θ_γ(κ)$ are
all equivalent for all limit ordinals $γ \leq κ$. $\mathcal{G}^θ_γ(κ,
ζ)$ is equivalent to $\mathcal{G}^θ_γ(κ)$ whenever $\mathrm{cof}(γ)
> ω$.
### Generalisations
(from {% cite Nielsen2018 %})
Now we can define $γ$-Ramsey cardinals for any ordinal $γ$ and other
variants: Let $κ$ be a cardinal and $γ \leq κ$ an ordinal:
- $κ$ is **$γ$-Ramsey** iff player I does not have a winning strategy
in $\mathcal{G}^θ_γ(κ)$ for all regular $θ > κ$.
- $κ$ is **strategic $γ$-Ramsey** iff player II does have a winning
strategy in $\mathcal{G}^θ_γ(κ)$ for all regular $θ > κ$.
- **(Strategic) genuine $γ$-Ramseys** and **(strategic) normal
$γ$-Ramseys** are defined analogously, with the additional
requirement for the last measure $μ_γ$ to be genuine and normal,
respectively.
- $κ$ is lt;γ$-Ramsey iff it is $α$-Ramsey for every $α < γ$.
- $κ$ is **almost fully Ramsey** iff it is lt;κ$-Ramsey.
- $κ$ is **fully Ramsey** iff it is $κ$-Ramsey.
- $κ$ is **coherent lt;γ$-Ramsey** iff it is strategic
lt;γ$-Ramsey and a single strategy works for player II in
$\mathcal{G}_α(κ)$ for every $α < γ$.
- I.e., in a choice of strategies for each $α$, strategies for
greater $α$ contain strategies for lesser $α$. Full definition
in {% cite Nielsen2018 %}.
(Some of the notions defined in
{% cite Nielsen2018 %} were not new, but
gained more convenient names.)
### Filter property
(from {% cite Holy2018 %})
$κ$ has the **filter property** iff for every subset $X$ of
$\mathcal{P}(κ)$ of size $≤κ$, there is a lt;κ$-complete filter $F$
on $κ$ which measures $X$. For normal filter we talk about *normal
filter property*.
Strengthenings:
- $κ$ has the **$γ$-filter property** iff player I does not have a
winning strategy in $G^θ_γ(κ)$.
- $κ$ has the **strategic $γ$-filter property** iff player II does
have a winning strategy in $G^θ_γ(κ)$.
- $κ$ has the **well-founded $(γ, θ)$-filter property** iff player I
does not have a winning strategy in $wfG^θ_γ(κ)$.
- $κ$ has the **well-founded $γ$-filter property** iff it has the
well-founded $(γ, θ)$-filter property for all regular $θ > κ$.
For $γ_1 > γ_0$, the $γ_1$-filter property implies the
$γ_0$-filter property.
### Results
Results in the finite case (for $n <
ω$):{% cite Nielsen2018 %}
- For a cardinal $κ=κ^{<κ}$
- $κ$ is [[weakly compact]]
iff it is 0-Ramsey;
- $κ$ is
<a href="Weakly_ineffable" class="mw-redirect" title="Weakly ineffable">weakly ineffable</a>
iff it is genuine 0-Ramsey;
- $κ$ is
[ineffable](Ineffable.md "Ineffable")
iff it is normal 0-Ramsey. (An uncountable cardinal κ has the
normal filter property iff it is
ineffable.{% cite Holy2018 %})
- Every $n$-Ramsey $κ$ is $Π^1_{2 n+1}$-indescribable. This is
optimal, as $n$-Ramseyness can be described by a $Π^1_{2
n+2}$-formula.
- Every lt;ω$-Ramsey cardinal is $∆^2_0$-indescribable.
- Every normal $n$-Ramsey $κ$ is $Π^1_{2 n+2}$-indescribable. This is
optimal, as normal $n$-Ramseyness can be described by a $Π^1_{2
n+3}$-formula.
- Every $n+1$-Ramsey is a normal $n$-Ramsey stationary limit of normal
$n$-Ramseys and every normal $n$-Ramsey is an $n$-Ramsey stationary
limit of $n$-Ramseys.
- Genuine- and normal $n$-Ramseys are downwards absolute to $L$.
- Every $(n+1)$-Ramsey is normal $n$-Ramsey in $L$. Therefore,
lt;ω$-Ramseys are downwards absolute to $L$.
Results for
$ω$-Ramsey:{% cite Holy2018 %}
- $ω$-Ramsey cardinals are super weakly Ramsey limits of super weakly
Ramsey cardinals.
- $ω$-Ramsey cardinals are limits of cardinals with the $ω$-filter
property (=completely
[ineffable](Ineffable.md "Ineffable"){% cite Nielsen2018 %}).
- $ω$-Ramsey cardinals are downwards absolute to $L$. If
<a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$0^♯lt;/a>
exists, then all Silver indiscernibles are $ω$-Ramsey in $L$.
Results for strategic
$ω$-Ramsey:{% cite Nielsen2018 %}
- <a href="Virtually_measurable" class="mw-redirect" title="Virtually measurable">Virtually measurable</a>
cardinals, strategic $ω$-Ramsey cardinals and
[remarkable](Remarkable.md "Remarkable")
cardinals are equiconsistent.
- Every virtually measurable cardinal is strategic $ω$-Ramsey, and
every strategic $ω$-Ramsey cardinal is virtually measurable in
$L$.
- If $κ$ is virtually measurable, then either $κ$ is
[remarkable](Remarkable.md "Remarkable")
in $L$ or $L_κ \models \text{“there is a proper class of
virtually measurables”}$.
- Remarkable cardinals are strategic $ω$-Ramsey limits of
$ω$-Ramsey cardinals.
- Therefore, if $κ$ is a strategic ω-Ramsey cardinal then $L_κ
\models \text{“there is a proper class of $ω$-Ramseys”}$.
Equiconsistency with the
[measurable](Measurable.md "Measurable")
cardinal:
- The existence of a strategic $(ω+1)$-Ramsey cardinal (and of
strategic fully Ramsey cardinal) is equiconsistent with the
existence of a measurable
cardinal.{% cite Nielsen2018 %}
- If $κ$ is a measurable cardinal, then $κ$ is $κ$-very Ramsey. If a
cardinal is $ω_1$-very Ramsey (=strategic $ω_1$-Ramsey cardinal),
then it is measurable in the [](Core%20models.md)
unless
<a href="Zero_pistol" class="mw-redirect" title="Zero pistol">$0^\Plt;/a>
exists and an inner model with a
[Woodin](Woodin.md "Woodin")
cardinal exists. {% cite Sharpe2011 Nielsen2018 %}
- If $κ$ is uncountable, $κ = κ^{<κ}$ and $2^κ = κ^+$, then the
following are
equivalent:{% cite Holy2018 %}
- $κ$ is measurable.
- $κ$ satisfies the $κ^+$-filter property.
- $κ$ satisfies the strategic $κ^+$-filter property.
- On the other hand, starting from a $κ^{++}$-tall cardinal $κ$, it is
consistent that there is a cardinal $κ$ with the strategic
$κ^+$-filter property, however $κ$ is not measurable.
Being downwards absolute to the [](Core%20models.md):{% cite Nielsen2018 %}
- If
<a href="Zero_pistol" class="mw-redirect" title="Zero pistol">$0^\Plt;/a>
does not exist:
- If $α$ is a limit ordinal with uncountable cofinality, then
being $α$-Ramsey is downwards absolute to $\mathbf{K}$.
- If $α$ is a limit ordinal, then genuine $α$-Ramseyness and
normal $α$-Ramseyness are both downwards absolute to
$\mathbf{K}$.
- if $α$ is a limit of limit ordinals, then lt;α$-Ramseyness is
downwards absolute to $\mathbf{K}$.
Strategic $α$-Ramsey (including coherent lt;α$-Ramsey) and $α$-very
Ramsey:{% cite Nielsen2018 %}
- For limit ordinal $α$, every coherent lt;ωα$-Ramsey is $ωα$-very
Ramsey.
- For any ordinal $α$, every coherent lt;α$-very Ramsey is coherent
lt;α$-Ramsey.
- For limit ordinal $α$, $κ$ is $ωα$-very Ramsey iff it is coherent
lt;ωα$-Ramsey.
- $κ$ is $λ$-very Ramsey iff it is strategic $λ$-Ramsey for any $λ$
with uncountable cofinality.
Hierarchy:{% cite Holy2018 %}
- If $ω ≤ α_0 < α_1 ≤ κ$, both $α_0$ and $α_1$ are cardinals,
and $κ$ is $α_1$-Ramsey, then there is a proper class of
$α_0$-Ramsey cardinals in $V_κ$. If $α_0$ is regular, then $κ$ is
a limit of $α_0$-Ramsey cardinals.
- If $α ≤ κ$ are both cardinals and $κ$ is $α$-Ramsey, then $κ$ has a
well-founded $α$-filter sequence.
- If $ω ≤ α < β ≤ κ$ are cardinals and $κ$ has a $β$-filter
sequence, then there is a proper class of $α$-Ramsey cardinals in
$V_κ$. If $α$ is regular, then $κ$ is a limit of $α$-Ramsey
cardinals.
Other:
- Every $(ω+1)$-Ramsey cardinal is a completely Ramsey stationary
limit of completely Ramsey
cardinals.{% cite Nielsen2018 %}
- Every strongly Ramsey cardinal is a stationary limit of almost fully
Ramseys.{% cite Nielsen2018 %}
- Fully Ramsey cardinals are super Ramsey limits of super Ramsey
cardinals.{% cite Holy2018 %}
- [Measurable](Measurable.md "Measurable")
cardinals are limits of fully Ramsey
cardinals.{% cite Holy2018 %}
### $(α, β)$-Ramsey cardinals
(All information from {% cite Nielsen2018 %})
$κ$ is **$(α, β)$-Ramsey** iff player I has no winning strategy in
$\mathcal{G}^θ_α(κ, β)$ for all regular $θ > κ$.
Of course, this notion is interesting only for $\mathrm{cof}(α) = ω$.
$α$-Ramsey cardinals are by definition equivalent to $(α, 0)$-Ramsey
cardinals.
Position in the hierarchy of
[Erdős](Erdos.md "Erdos") and
iterable cardinals:
- For $β > 0$, every $(α, β)$-Ramsey is a $β$-iterable stationary
limit of $β$-iterables.
- For additively closed $ω \leq α \leq ω_1$, any $α$-Erdős cardinal
is a limit of $(ω, α)$-Ramsey cardinals.
This means also that $(ω, α)$-Ramsey cardinals are consistent with $V =
L$ if $α < ω_1^L$ and that they are not if $α = ω_1$ .
### $(γ, θ)$-Ramsey cardinals
$κ$ is **$(γ, θ)$-Ramsey** iff player I has no winning strategy in
$\mathcal{G}^θ_γ(κ)$ (i.e. $κ$ is $γ$-Ramsey iff it is $(γ, θ)$-Ramsey
for every $θ > κ$). Not much is known about them in
general.{% cite Nielsen2018 %}
## M-rank
(from {% cite Carmody2016 %})
M-rank for Ramsey and Ramsey-like cardinals is analogous to [](Mitchell%20rank%20and%20Mitchell%20order.md).
A difference is that M-rank for Ramsey-like cardinals can be at most
$\kappa^+$ (because an ultrapower of a weak $κ$-model has size at most
$κ$) and Mitchell rank for
[measurable](Measurable.md "Measurable")
cardinals can be at most $(2^\kappa)^+$.
Definition of the M-order: For $κ$ having a large-cardinal property
$\mathscr{P}$ with an embedding characterisation and for two witness
collections $\mathcal{U}$ and $\mathcal{W}$ of
$\mathscr{P}$-measures, we say that $U⊳W$ if
- for every $W∈\mathcal{W}$ and $A⊆κ$ in the ultrapower $N_W$ of
$M_W$ by $W$, there is an $A$-good $U∈ \mathcal{U} ∩ N_W$ such
that $N_W \models \text{“$\mathcal{U}$ is an $A$-good
$\mathscr{P}$-measure on $κ$”}$.
- $\mathcal{U} ⊆ ⋃_{W∈\mathcal{W}} N_W$.
Results:
- Any strongly Ramsey cardinal $κ$ has Ramsey M-rank $κ^+$,
- any super Ramsey cardinal $κ$ has strongly Ramsey M-rank $κ^+$
- and any measurable cardinal $κ$ has super Ramsey M-rank $κ^+$.
Ramsey and Ramsey-like M-orders can be softly killed (Rank $α$ can be
turned into rank $β$ for any $β < α$) using cofinality-preserving
forcing extension.