# Difference between revisions of "Ramsey"

Ramsey cardinals were introduced by Erdős and Hajnal in [1]. 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 $\mathcal A\models\varphi (\overline \alpha)\leftrightarrow \varphi(\overline \beta)$. [2]

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 $\langle L_\kappa[A], A\rangle$. [3]

$M$-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 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. [4] (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 $\kappa$. [3]

## Ramsey cardinals and the constructible universe

Ramsey cardinals imply that $0^\sharp$ exists and hence there cannot be Ramsey cardinals in $L$. [4]

## Relations with other large cardinals

• Measurable cardinals are Ramsey and stationary limits of Ramsey cardinals. [1]
• 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.[5]
• There are stationarily many completely ineffable, greatly Erdős cardinals below any Ramsey cardinal.[6]

## Ramsey cardinals and forcing

• Ramsey cardinals are preserved by small forcing. [4]
• 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. [7]
• If $\kappa$ is Ramsey, there is a forcing extension in which $\kappa$ remains Ramsey and $2^\kappa\gt\kappa$. Indeed, 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 $\delta$. [8]
• 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. [7]

## Ramsey-like cardinals

There are many Ramsey-like cardinals. [5, 6, 9, 10, 11]

### Completely Romsey cardinals etc.

(All information in this subsection are from [9] 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 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 $(2^κ)^+$ 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 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 subsubsection compares (Sharpe&Welch, 2011) and (Feng, 1990))

$Π_α$-Ramsey cardinals correspond to $α$-Ramsey and $α$-Ramsey$^s$ in [6].[10] (The “$^s$” stands for “stationary”.[6])

$Π_{2 n}$-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

• 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 model.
• Any measurable cardinal is super completely Ramsey and a stationary limit of super completely Ramsey cardinals.
• 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.
• 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 $κ$.
• If $κ$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.[10]
• 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.).[6]

### Super Ramsey cardinal

Super Ramsey cardinals were introduced by Gitman in [5] (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.
• Super Ramsey cardinals are strongly Ramsey limits of strongly Ramsey cardinals.
• Super Ramseyness is downward absolute to $K$. [12]
• The required $M$ for a super Ramsey embedding is stationarily correct.
• The consistency strength of a super Ramsey cardinal is stronger than that of a strongly Ramsey cardinal and weaker than that of a measurable cardinal.

### Strongly Ramsey cardinal

Strongly Ramsey cardinals were introduced by Gitman in [5] (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 $\kappa$.

• Measurable cardinals are strongly Ramsey limits of strongly Ramsey cardinals.
• 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. [7]
• The consistency strength of strongly Ramsey cardinals is stronger than that of Ramsey cardinals.
• Strong Ramseyness is downward absolute to $K$. [12]
• Strongly Ramsey cardinals are limits of completely Ramsey cardinals, but are not necessarily completely Ramsey themselves.[12]

### Mahlo–Ramsey cardinals

The property of being Mahlo–Ramsey (MR) is a slight strengthening of Ramseyness introduced in analogy to Mahlo cardinals in [6] (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

(all information from [6])

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 $G_r(X, α)$.

For $n < ω$, the games $G_R(X, n)$ and $G_r(X, n)$ coincide.

Strength:

• Regular uncountable cardinal $κ$ is completely Ramsey iff $κ$ is $ω$-very Ramsey.
• If $κ$ is a measurable cardinal, then $κ$ is $κ$-very Ramsey.
• If a cardinal is $ω_1$-very Ramsey, then it is measurable in the core model or $0^\P$ exists.

It is not known how strong $α$-very Ramsey is for countable $α$.

### Virtually Ramsey cardinal

Virtually Ramsey cardinals were introduced by Sharpe and Welch in [6]. 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 [6]. 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. It is consistent from a strongly Ramsey cardinal that there is a virtually Ramsey cardinal that is not Ramsey. It is open whether such separation is possible from just a Ramsey cardinal and whether virtually Ramsey cardinals are weaker than Ramsey cardinals.[12]

If κ is virtually Ramsey then κ is greatly Erdős.[6]

### Almost Ramsey cardinal

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 $\eta_\alpha<\kappa$.
• $\kappa=\text{sup}\{\eta_\alpha:\alpha<\kappa\}$

Where $\eta_\alpha$ 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.[6]

### Super weakly Ramsey cardinal

(All from [10])

A cardinal $κ$ is super weakly Ramsey iff every $A ⊆ κ$ is contained, as an element, in a weak κ-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 limits of weakly Ramsey cardinals.
• Super weakly Ramsey cardinals are ineffable.

### $α$-Ramsey cardinal

(All from [10] unless otherwise noted)

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 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).

Strength:

• $ω$-Ramsey cardinals are super weakly Ramsey limits of super weakly Ramsey cardinals.
• $ω_1$-Ramsey cardinals are completely Ramsey limits of completely Ramsey cardinals.
• If $κ$ is $κ$-Ramsey, then $κ$ is a super Ramsey limit of super Ramsey cardinals.
• 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.
• Measurable cardinals are limits of regular cardinals $α$ which are $α$-Ramsey.
• 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.
• $ω$-Ramsey cardinals are limits of cardinals with the $ω$-filter property (=completely ineffable[11]).
• $ω$-Ramsey cardinals are downwards absolute to $L$. If $0^♯$ exists, then all Silver indiscernibles are $ω$-Ramsey in $L$.

#### Variants

(all information from [11])

Strategic $α$-Ramsey cardinals, genuine $α$-Ramsey cardinals, normal $α$-Ramsey cardinals

......

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.

### $\alpha$-iterable cardinal

The $\alpha$-iterable cardinals for $1\leq\alpha\leq\omega_1$ were introduced by Gitman in [12]. They form a hierarchy of large cardinal notions strengthening weakly compact cardinals, while weakening the $M$-ultrafilters 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 [13], 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.

• $1$-iterable cardinals are weakly ineffable and stationary limits of completely ineffable cardinals. The least $1$-iterable cardinal is not ineffable. [5]
• An $\alpha$-iterable cardinal is $\beta$-iterable and a stationary limit of $\beta$-iterable cardinals for every $\beta<\alpha$. [12]
• 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. [6]
• It is consistent from an $\omega$-Erdős cardinal that for every $n\in\omega$, there is a proper class of $n$-iterable cardinals.
• A $2$-iterable cardinal is a limit of remarkable cardinals. [12]
• A remarkable cardinal implies the consistency of a $1$-iterable cardinal. [12]
• $\omega_1$-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 $\alpha$. [12]
• $\alpha$-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 $2^\kappa\gt\kappa$. [14]
• 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.[15]
• If $κ$ is 2-iterable, then $V_κ$ is a model of proper class many virtually $C^{(n)}$-extendible cardinals for every $n < ω$ and of proper class many virtually Shelah for supercompactness cardinals.[15]
• Virtually extendible cardinals are 1-iterable limits of 1-iterable cardinals.[15]
• 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.[15]
• Every element of a club $C$ witnessing that $κ$ is a Silver cardinal is virtually rank-into-rank.[15]
• 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 Silver cardinals), then each $ξ ∈ C$ is $< ω_1$-iterable.[15]

## M-rank

M-rank for Ramsey and Ramsey-like cardinals is analogous to Mitchell rank. A difference is that M-rank for Ramsey-like cardinals can be at most $\kappa^+$ and Mitchell rank for measurable cardinals can be at most $(2^\kappa)^+$. 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 $κ^+$.[16]

## References

1. Erdős, Paul and Hajnal, Andras. Some remarks concerning our paper On the structure of set-mappings''. Non-existence of a two-valued $\sigma$-measure for the first uncountable inaccessible cardinal. Acta Math Acad Sci Hungar 13:223--226, 1962. MR   bibtex
2. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www   bibtex
3. Dodd, Anthony and Jensen, Ronald. The core model. Ann Math Logic 20(1):43--75, 1981. www   DOI   MR   bibtex
4. Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www   bibtex
5. Gitman, Victoria. Ramsey-like cardinals. The Journal of Symbolic Logic 76(2):519-540, 2011. www   arχiv   MR   bibtex
6. Sharpe, Ian and Welch, Philip. Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Ann Pure Appl Logic 162(11):863--902, 2011. www   DOI   MR   bibtex
7. Gitman, Victoria and Johnstone, Thomas A. Indestructibility for Ramsey and Ramsey-like cardinals. (In preparation) www   bibtex
8. Cody, Brent and Gitman, Victoria. Easton's theorem for Ramsey and strongly Ramsey cardinals. Annals of Pure and Applied Logic 166(9):934 - 952, 2015. www   DOI   bibtex
9. Feng, Qi. A hierarchy of Ramsey cardinals. Annals of Pure and Applied Logic 49(3):257 - 277, 1990. DOI   bibtex
10. Holy, Peter and Schlicht, Philipp. A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae 242:49-74, 2018. www   arχiv   DOI   bibtex
11. Nielsen, Dan Saattrup and Welch, Philip. Games and Ramsey-like cardinals. , 2018. arχiv   bibtex
12. Gitman, Victoria and Welch, Philip. Ramsey-like cardinals II. J Symbolic Logic 76(2):541--560, 2011. www   arχiv   MR   bibtex
13. Gaifman, Haim. Elementary embeddings of models of set-theory and certain subtheories. Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967), pp. 33--101, Providence R.I., 1974. MR   bibtex
14. Gitman, Victoria and Shindler, Ralf. Virtual large cardinals. www   bibtex
15. Carmody, Erin and Gitman, Victoria and Habič, Miha E. A Mitchell-like order for Ramsey and Ramsey-like cardinals. , 2016. arχiv   bibtex
Main library