Ramsey cardinal
Ramsey cardinals were introduced by Erdős and Hajnal in
(Erdős & Hajnal, 1962). Their consistency strength lies
strictly between $0^\sharp$ and measurable cardinals.
There are many Ramseylike cardinals with strength between weakly
compact and measurable cardinals inclusively.
(Feng, 1990; Gitman, 2011; Sharpe & Welch, 2011; Holy & Schlicht, 2018; Nielsen & Welch, 2018)
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
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 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
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.
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 $\mathcal A\models\varphi (\overline
\alpha)\leftrightarrow \varphi(\overline \beta)$.
(Jech, 2003)
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 $\langle L_\kappa[A], A\rangle$.
(Dodd & Jensen, 1981)
$M$ultrafilters:
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 nonempty
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 wellfounded. If the
$M$ultrafilter is moreover weakly amenable, then a weakly amenable
ultrafilter on the image of $\kappa$ in the wellfounded 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
wellfounded model. (Kanamori, 2009) (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 $\kappa$.
(Dodd & Jensen, 1981)
Ramsey cardinals and the constructible universe
Ramsey cardinals imply that
$0^\sharp$
exists and hence there cannot be Ramsey cardinals in $L$.
(Kanamori, 2009)
Relations with other large cardinals

Measurable
cardinals are Ramsey and stationary limits of Ramsey cardinals.
(Erdős & Hajnal, 1962)
 Ramsey cardinals are
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
completely ineffable
cardinals, they are
weakly ineffable,
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.(Gitman, 2011)
 There are stationarily many completely ineffable, greatly
Erdős
cardinals below any Ramsey
cardinal.(Sharpe & Welch, 2011)
Weaker Ramseylike:
 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
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
cardinal. (Sharpe & Welch, 2011)
 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. (Gitman & Welch, 2011; Gitman & Johnstone, n.d.)
Stronger Ramseylike:
 If $κ$ is $Π_1$Ramsey, then the set of Ramsey cardinals less then
$κ$ is in the $Π_1$Ramsey filter on
$κ$.(Feng, 1990)
 Strongly Ramsey cardinals are Ramsey and stationary limits of Ramsey
cardinals.(Gitman, 2011)
 Mahlo–Ramsey cardinals are a direct strengthening of
Ramseyness.(Sharpe & Welch, 2011)
Ramsey cardinals and forcing
 Ramsey cardinals are preserved by small forcing.
(Kanamori, 2009)
 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.
(Gitman & Johnstone, n.d.)
 If $\kappa$ is Ramsey, there is a forcing extension in which
$\kappa$ remains Ramsey and
$2^\kappa\gt\kappa$. (Gitman & Johnstone, n.d.; Cody & Gitman, 2015)
 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 $\delta$.
(Cody & Gitman, 2015)
 There is a forcing extension in which $κ$ is the first cardinal
at which the $\mathrm{GCH}$ fails.
(Gitman & Johnstone, n.d.)
 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.
(Gitman & Johnstone, n.d.)
Completely Romsey cardinals etc.
(All information in this section are from
(Feng, 1990) 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
ideal
on $κ$, its dual $\mathcal{R}^*(\mathcal{P}_{<κ}(κ))$ the Ramsey
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 nonstationary subsets of $κ$)
 for $n < ω$, $I_n^κ = \mathcal{R}(I_{n2}^κ)$
 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 $(2^κ)^+$ for
every regular uncountable $κ$ — even if the ideals are trivial)
$α$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 0hyper 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
(Sharpe & Welch, 2011; Holy & Schlicht, 2018)
(The “$^s$” stands for
“stationary”.(Sharpe & Welch, 2011))
$Π_{2 n}$Ramsey cardinals are SharpeWelch $n$Ramsey and $Π_{2 n +
1}$Ramsey cardinals are $n$Ramsey$^s$.
For infinite $α$, $Π_α$Ramsey, SharpeWelch $α$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 DoddJensen core
model.
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
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.
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.).(Sharpe & Welch, 2011)
 If $κ$ is completely Ramsey, then $κ$ is
$Π_1^2$indescribable.(Holy & Schlicht, 2018)
Equivalence:
Relation with other variants of Ramseyness:
 Strongly Ramsey cardinals are limits of completely Ramsey cardinals,
but are not necessarily completely Ramsey
themselves.(Gitman & Welch, 2011)
 Every $(ω+1)$Ramsey cardinal is a completely Ramsey stationary
limit of completely Ramsey
cardinals.(Nielsen & Welch, 2018)
 Any $\Pi_2$Ramsey cardinal is $α$Mahlo–Ramsey for all $α <
κ^+$. (Sharpe & Welch, 2011)
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
$<\kappa$, there is are sets of indiscernibles
$I\subseteq\kappa$ for $\mathcal{M}$ of any size $<\kappa$.
 For every $\alpha<\kappa$, $\eta_\alpha$ exists and
$\eta_\alpha<\kappa$.
 $\kappa=\text{sup}\{\eta_\alpha:\alpha<\kappa\}$
($\eta_\alpha$ is the
$\alpha$Erdős
cardinal.)
Every almost Ramsey cardinal is a $\beth$fixed
point, but
the least almost Ramsey cardinal, if it exists, has cofinality
$\omega$. In fact, the least almost Ramsey cardinal is not
weakly inaccessible,
worldly,
or
correct.
However, if the least almost Ramsey cardinal exists, it is larger than
the least
$\omega_1$Erdős
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
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.(Sharpe & Welch, 2011)
Strongly Ramsey cardinal
Strongly Ramsey cardinals were introduced by Gitman in
(Gitman, 2011) (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.(Gitman & Welch, 2011)
 Every strongly Ramsey cardinal is a stationary limit of almost fully
Ramseys.(Nielsen & Welch, 2018)
 Strongly Ramsey cardinals are Ramsey and stationary limits of Ramsey
cardinals.
 The least strongly Ramsey cardinal is not
ineffable.
 Forcing related properties of strongly Ramsey cardinals are the same
as those of Ramsey cardinals described above.
(Gitman & Johnstone, n.d.)
 Strong Ramseyness is downward absolute to $K$.
(Gitman & Welch, 2011)
Super Ramsey cardinal
Super Ramsey cardinals were introduced by Gitman in
(Gitman, 2011) (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
cardinals are super Ramsey limits of super Ramsey cardinals.
 Fully Ramsey cardinals are super Ramsey limits of super Ramsey
cardinals.(Holy & Schlicht, 2018)
 Super Ramsey cardinals are strongly Ramsey limits of strongly Ramsey
cardinals.
 Super Ramseyness is downward absolute to $K$.
(Gitman & Welch, 2011)
 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
(Gitman & Welch, 2011). 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 wellfounded,
 $U$ is 1good if it is 0good and weakly amenable,
 for an ordinal $\alpha>1$, $U$ is $\alpha$good, if it produces
at least $\alpha$many wellfounded iterated ultrapowers.
Using a theorem of Gaifman
(Gaifman, 1974), 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
weakly ineffable
and stationary limits of
completely ineffable
cardinals. The least $1$iterable cardinal is not ineffable.
(Gitman, 2011)
 Super weakly Ramsey cardinals are weakly Ramsey (=$1$iterable)
limits of weakly Ramsey cardinals.
Upper limit:
Hierarchy:
 An $\alpha$iterable cardinal is $\beta$iterable and a stationary
limit of $\beta$iterable cardinals for every $\beta<\alpha$.
(Gitman & Welch, 2011)
 For $β > 0$, every $(α, β)$Ramsey is a $β$iterable stationary
limit of $β$iterables.
 It is consistent from an
$\omega$Erdős
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.(Gitman & Shindler, n.d.)
 A virtually
$n$huge*
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.(Gitman & Shindler, n.d.)
 Every virtually rankintorank cardinal is an
$ω$iterable
limit of $ω$iterable
cardinals.(Gitman & Shindler, n.d.)
Between $1$ and $2$iterable:
Absoluteness:
 $\omega_1$iterable cardinals imply that
$0^\sharp$
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
$0^\sharp$
exists, then every Silver indiscernible is $\alpha$iterable in $L$
for every $L$countable $\alpha$.
(Gitman & Welch, 2011)
 $\alpha$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
$2^\kappa\gt\kappa$.
(Gitman & Johnstone, n.d.)
Mahlo–Ramsey cardinals
The property of being Mahlo–Ramsey (MR) is a slight strengthening of
Ramseyness introduced in analogy to
Mahlo
cardinals in (Sharpe & Welch, 2011) (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
SharpeWelch $\alpha$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
$G_r(X, α)$.(Sharpe & Welch, 2011)
For $n < ω$, the games $G_R(X, n)$ and $G_r(X, n)$
coincide.(Sharpe & Welch, 2011)
In analogy to coherent $<α$very Ramsey, one can define coherent
$<α$very Ramsey cardinals. $α$very Ramsey cardinals are equivalent
to coherent $<α$very Ramsey cardinals for limit $α$ and to
$<(α+1)$very Ramsey cardinals in general. (This just allows to
“subtract one” for successor
ordinals.)(Nielsen & Welch, 2018)
Results:
Additional results from
(Nielsen & Welch, 2018):
 For limit ordinal $α$, every coherent $<ωα$Ramsey is $ωα$very
Ramsey.
 For any ordinal $α$, every coherent $<α$very Ramsey is coherent
$<α$Ramsey.
 For limit ordinal $α$, $κ$ is $ωα$very Ramsey iff it is coherent
$<ωα$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
(Sharpe & Welch, 2011). 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
(Sharpe & Welch, 2011). 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 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. (Gitman & Welch, 2011; Gitman & Johnstone, n.d.)
If κ is virtually Ramsey then κ is greatly
Erdős.(Sharpe & Welch, 2011)
Super weakly Ramsey cardinal
(All from (Holy & Schlicht, 2018))
A cardinal $κ$ is super weakly Ramsey iff every $A ⊆ κ$ is
contained, as an element, in a weak
κmodel $M
≺ H(κ^+)$ 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.
 $ω$Ramsey cardinals are super weakly Ramsey limits of super weakly
Ramsey cardinals.
$α$Ramsey cardinal etc.
$α$Ramsey cardinal for cardinal $α$
(All from (Holy & Schlicht, 2018))
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
$<α$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
$<κ$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 wellfounded
$ω$filter property.
This characterisation works better for singular alpha $α$ (the original
one would imply being $α^+$Ramsey; wellfounded $α$filter property is
better for countable cofinality).
Games for definitions
(from (Nielsen & Welch, 2018) unless otherwise
noted)
For a weak
$κ$model
$\mathcal{M}$, $μ$ is an
$\mathcal{M}$measure
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 wellfounded filter
games(Holy & Schlicht, 2018)
$wfG^θ_γ(κ)$ (full definition in sources).
 Player I
(challenger(Holy & Schlicht, 2018))
gives $\subseteq$increasing $κ$models $\mathcal{M}_α ≺ H_θ$,
 player II
(judge(Holy & Schlicht, 2018)) 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 (Nielsen & Welch, 2018))
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 $<γ$Ramsey iff it is $α$Ramsey for every $α < γ$.
 $κ$ is almost fully Ramsey iff it is $<κ$Ramsey.
 $κ$ is fully Ramsey iff it is $κ$Ramsey.
 $κ$ is coherent $<γ$Ramsey iff it is strategic
$<γ$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 (Nielsen & Welch, 2018).
(Some of the notions defined in
(Nielsen & Welch, 2018) were not new, but
gained more convenient names.)
Filter property
(from (Holy & Schlicht, 2018))
$κ$ has the filter property iff for every subset $X$ of
$\mathcal{P}(κ)$ of size $≤κ$, there is a $<κ$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 wellfounded $(γ, θ)$filter property iff player I
does not have a winning strategy in $wfG^θ_γ(κ)$.
 $κ$ has the wellfounded $γ$filter property iff it has the
wellfounded $(γ, θ)$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 $n <
ω$):(Nielsen & Welch, 2018)
 For a cardinal $κ=κ^{<κ}$
 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 $<ω$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,
$<ω$Ramseys are downwards absolute to $L$.
Results for
$ω$Ramsey:(Holy & Schlicht, 2018)
 $ω$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(Nielsen & Welch, 2018)).
 $ω$Ramsey cardinals are downwards absolute to $L$. If
$0^♯$
exists, then all Silver indiscernibles are $ω$Ramsey in $L$.
Results for strategic
$ω$Ramsey:(Nielsen & Welch, 2018)

Virtually measurable
cardinals, strategic $ω$Ramsey cardinals and
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
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 ωRamsey cardinal then $L_κ
\models \text{“there is a proper class of $ω$Ramseys”}$.
Equiconsistency with the
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.(Nielsen & Welch, 2018)
 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
model
unless
$0^\P$
exists and an inner model with a
Woodin
cardinal exists. (Sharpe & Welch, 2011; Nielsen & Welch, 2018)
 If $κ$ is uncountable, $κ = κ^{<κ}$ and $2^κ = κ^+$, then the
following are
equivalent:(Holy & Schlicht, 2018)
 $κ$ 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
model:(Nielsen & Welch, 2018)
 If
$0^\P$
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 $<α$Ramseyness is
downwards absolute to $\mathbf{K}$.
Strategic $α$Ramsey (including coherent $<α$Ramsey) and $α$very
Ramsey:(Nielsen & Welch, 2018)
 For limit ordinal $α$, every coherent $<ωα$Ramsey is $ωα$very
Ramsey.
 For any ordinal $α$, every coherent $<α$very Ramsey is coherent
$<α$Ramsey.
 For limit ordinal $α$, $κ$ is $ωα$very Ramsey iff it is coherent
$<ωα$Ramsey.
 $κ$ is $λ$very Ramsey iff it is strategic $λ$Ramsey for any $λ$
with uncountable cofinality.
Hierarchy:(Holy & Schlicht, 2018)
 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
wellfounded $α$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:
$(α, β)$Ramsey cardinals
(All information from (Nielsen & Welch, 2018))
$κ$ 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 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.(Nielsen & Welch, 2018)
Mrank
(from (Carmody et al., 2016))
Mrank for Ramsey and Ramseylike cardinals is analogous to Mitchell
rank.
A difference is that Mrank for Ramseylike cardinals can be at most
$\kappa^+$ (because an ultrapower of a weak $κ$model has size at most
$κ$) and Mitchell rank for
measurable
cardinals can be at most $(2^\kappa)^+$.
Definition of the Morder: For $κ$ having a largecardinal 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 Mrank $κ^+$,
 any super Ramsey cardinal $κ$ has strongly Ramsey Mrank $κ^+$
 and any measurable cardinal $κ$ has super Ramsey Mrank $κ^+$.
Ramsey and Ramseylike Morders can be softly killed (Rank $α$ can be
turned into rank $β$ for any $β < α$) using cofinalitypreserving
forcing extension.
References
 Erdős, P., & Hajnal, A. (1962). Some remarks concerning our paper “On the structure of setmappings\’\’. Nonexistence of a twovalued σmeasure for the first uncountable inaccessible cardinal. Acta Math. Acad. Sci. Hungar., 13, 223–226.
 Feng, Q. (1990). A hierarchy of Ramsey cardinals. Annals of Pure and Applied Logic, 49(3), 257–277. https://doi.org/10.1016/01680072(90)90028Z
 Gitman, V. (2011). Ramseylike cardinals. The Journal of Symbolic Logic, 76(2), 519–540. http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf
 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. https://doi.org/10.1016/j.apal.2011.04.002
 Holy, P., & Schlicht, P. (2018). A hierarchy of Ramseylike cardinals. Fundamenta Mathematicae, 242, 49–74. https://doi.org/10.4064/fm39692017
 Nielsen, D. S., & Welch, P. (2018). Games and Ramseylike cardinals.
 Jech, T. J. (2003). Set Theory (Third). SpringerVerlag. https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf
 Dodd, A., & Jensen, R. (1981). The core model. Ann. Math. Logic, 20(1), 43–75. https://doi.org/10.1016/00034843(81)900115
 Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). SpringerVerlag. https://link.springer.com/book/10.1007%2F9783540888673
 Gitman, V., & Welch, P. (2011). Ramseylike cardinals II. J. Symbolic Logic, 76(2), 541–560. http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf
 Gitman, V., & Johnstone, T. A. Indestructibility for Ramsey and Ramseylike cardinals. https://victoriagitman.github.io/files/indestructibleramseycardinalsnew.pdf
 Cody, B., & Gitman, V. (2015). Eastonś theorem for Ramsey and strongly Ramsey cardinals. Annals of Pure and Applied Logic, 166(9), 934–952. https://doi.org/10.1016/j.apal.2015.04.006
 Gaifman, H. (1974). Elementary embeddings of models of settheory and certain subtheories. In Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967) (pp. 33–101). Amer. Math. Soc.
 Gitman, V., & Shindler, R. Virtual large cardinals. https://ivv5hpp.unimuenster.de/u/rds/virtualLargeCardinalsEdited5.pdf
 Bagaria, J., Gitman, V., & Schindler, R. (2017). Generic Vopěnkaś Principle, remarkable cardinals, and the weak Proper Forcing Axiom. Arch. Math. Logic, 56(12), 1–20. https://doi.org/10.1007/s001530160511x
 Carmody, E., Gitman, V., & Habič, M. E. (2016). A Mitchelllike order for Ramsey and Ramseylike cardinals.
Main library