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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 Ramsey-like 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


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 $\gamma\in I$,

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 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. (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

Weaker Ramsey-like:

Stronger Ramsey-like:

Ramsey cardinals and forcing

Completely Romsey cardinals etc.

(All information in this section are from (Feng, 1990) unless otherwise noted)

Basic definitions

$Π_α$-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

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

Note four facts:

For a regular uncountable cardinal $κ$, let $\vec f = ⟨f_α:α<κ^+⟩$ be the canonical sequence on $κ$.


(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 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.




Upper limit of consistency strength:



Relation with other variants of Ramseyness:

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:

($\eta_\alpha$ 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 $\kappa$.


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:

$\alpha$-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:

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.


Lower limit:

Upper limit:


Between $1$- and $2$-iterable:


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 $⟨ f_α
α < κ^+ ⟩$

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 $G_r(X, α)$.(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)


Additional results from (Nielsen & Welch, 2018):

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 κ-model $M ≺ H(κ^+)$ for which there exists a $κ$-powerset preserving elementary embedding $j∶ M → N$.


$α$-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:

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 (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 well-founded filter games(Holy & Schlicht, 2018) $wfG^θ_γ(κ)$ (full definition in sources).

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).

For convenience

$\mathcal{G}^θ_γ(κ)$, $\mathcal{G}^θ_γ(κ, 1)$ 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 $\mathrm{cof}(γ) > ω$.


(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:

(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.


For $γ_1 > γ_0$, the $γ_1$-filter property implies the $γ_0$-filter property.


Results in the finite case (for $n < ω$):(Nielsen & Welch, 2018)

Results for $ω$-Ramsey:(Holy & Schlicht, 2018)

Results for strategic $ω$-Ramsey:(Nielsen & Welch, 2018)

Equiconsistency with the measurable cardinal:

Being downwards absolute to the core model:(Nielsen & Welch, 2018)

Strategic $α$-Ramsey (including coherent $<α$-Ramsey) and $α$-very Ramsey:(Nielsen & Welch, 2018)

Hierarchy:(Holy & Schlicht, 2018)


$(α, β)$-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:

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)


(from (Carmody et al., 2016))

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 $\kappa^+$ (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 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


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.


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  12. Cody, B., & Gitman, V. (2015). Eastonś theorem for Ramsey and strongly Ramsey cardinals. Annals of Pure and Applied Logic, 166(9), 934–952.
  13. Gaifman, H. (1974). Elementary embeddings of models of set-theory 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.
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  16. Carmody, E., Gitman, V., & Habič, M. E. (2016). A Mitchell-like order for Ramsey and Ramsey-like cardinals.
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