# 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 $\kappa\rightarrow (\kappa)^{\lt\omega}_2$. A 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. 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 $n$. 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.

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.

A cardinal $\kappa$ is Ramsey if and only if 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 hence strongly unfoldable in $L$.
• Ramsey cardinals are stationary limits of ineffable cardinals, they are weakly ineffable but not necessarily ineffable.[5]

## Ramsey cardinals and forcing

• Ramsey cardinals are preserved by small forcing. [4]
• Ramsey cardinals are preserved by the canonical forcing of the ${\rm GCH}$ and by fast function forcing. [6]
• 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 $\delta$. [7]
• 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. [6]

## Strongly Ramsey Cardinals

Strongly Ramsey cardinals were introduced by Gitman in [5]. They strengthen the $M$-ultrafilters characterization of Ramsey cardinals from weak $\kappa$-models to $\kappa$-models. A weak $\kappa$-model $M$ is a $\kappa$-model if additionally $M^{\lt\kappa}\subseteq M$. 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 $\langle M,U\rangle$ satisifes that it is $\kappa$-complete and it must be correct about this since $M$ is closed under sequences of length less than $\kappa$.

• Measurable cardinas are strongly Ramsey.
• Strongly Ramsey cardinals are Ramsey and stationary limits of Ramsey cardinals.
• Strongly Ramsey cardinals are not necessarily ineffable. [5]
• Forcing related properties of strongly Ramsey cardinals are the same as those of Ramsey cardinals described above. [6]

## Virtually Ramsey Cardinals

Virtually Ramsey cardinals were introduced by Welch in [8]. 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 [8]. 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 Ramsey cardinal that there is a virtually Ramsey cardinal that is not Ramsey. It is open whether virtually Ramsey cardinals are weaker than Ramsey cardinals. [9]

## $\alpha$-iterable cardinals

The $\alpha$-iterable cardinals for $1\leq\alpha\leq\omega_1$ were introduced by Gitman in [5, 10, 11, 12, 13, 13, 13, 13, 13, 14, 15, 15, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 22, 23, 24, 25, 25, 25, 25, 26, 26, 26, 26, 26, 26, 26, 27, 28, 29, 29, 30, 31, 32, 32, 32, 32, 32, 33, 33, 33, 33, 33, 33, 33, 33, 33, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 35, 36, 36, 37, 37, 38, 39, 39, 40, 40, 41, 41, 41, 42, 42, 42, 43, 43, 43, 44, 44, 44, 45, 45, 46, 46, 47, 47, 47, 47, 48, 48, 48, 49, 49, 50, 51, 52, 53, 54, 54, 55, 56, 57, 58, 59, 59, 60, 60, 60, 60, 61, 61, 62, 63, 64, 65, 65, 65, 66, 66, 67, 68, 68, 69, 70, 71, 72, 72, 72, 72, 72, 72, 73, 73, 74, 75, 76, 77, 78, 79, 80, 80, 81, 82, 83]