This is a talk at the CUNY Set Theory Seminar, October 20, 2017.

Peter Holy and Philipp Schlicht recently introduced a robust hierarchy of Ramsey-like cardinals $\kappa$ using games in which player I plays an increasing sequence of $\kappa$-models and player II responds by playing an increasing sequence of $M$-ultrafilters for some cardinal $\alpha\leq\kappa$ many steps, with player II winning if she is able to continue finding the required filters [1]. The entire hierarchy sits below a measurable cardinal and intertwines with Ramsey cardinals, as well as the Ramsey-like cardinals I introduced in [2]. The cardinals in the hierarchy can also be defined by the existence of the kinds of elementary embeddings characterizing Ramsey cardinals and other cardinals in that neighborhood. Before getting to their hierarchy and the filter games, we need some background.

Large cardinals $\kappa$ below a measurable cardinal tend to be characterized by the existence of certain elementary embeddings of weak $\kappa$-models or $\kappa$-models. A *weak $\kappa$-model* is a transitive model of ${\rm ZFC}^-$ of size $\kappa$ and height above $\kappa$, which we should think of as a mini-universe of set theory; a $\kappa$-model is additionally closed under $\lt\kappa$-sequences. Given a weak $\kappa$-model $M$, we call $U\subseteq P(\kappa)\cap M$ an $M$-*ultrafilter* if the structure $\langle M,\in, U\rangle$ satisfies that $U$ is a normal ultrafilter. (Note that since an $M$-ultrafilter is only $\lt\kappa$-complete for sequences from $M$, the ultrapower by it need not be well-founded.) Obviously, if the ultrapower of a weak $\kappa$-model $M$ by an $M$-ultrafilter on $\kappa$ is well-founded, then we get an elementary embedding of $M$ into a transitive model $N$, and conversely if there is an elementary embedding $j:M\to N$ with $N$ transitive and critical point $\kappa$, then $U=\{A\in M\mid A\subseteq\kappa\text{ and }\kappa\in j(A)\}$ is an $M$-ultrafilter with a well-founded ultrapower. These types of elementary embeddings characterize, for instance, weakly compact cardinals. If $\kappa^{\lt\kappa}=\kappa$, then $\kappa$ is weakly compact whenever every $\kappa$-model has an $M$-ultrafilter on $\kappa$ (and hence a well-founded ultrapower).

An $M$-ultrafilter $U$, for a weak $\kappa$-model $M$, is called *weakly amenable* if for every $X\in M$, which $M$ thinks has size $\kappa$, $X\cap U\in M$. Because a weakly amenable $M$-ultrafilter is partially internal to $M$, we are able to define its iterates and iterate the ultrapower construction as we would do with a measure on $\kappa$. If $j:M\to N$ is the ultrapower by a weakly amenable $M$-ultrafilter on $\kappa$, then $M$ and $N$ have the same subsets of $\kappa$, and conversely if $M$ and $N$ have the same subsets of $\kappa$ and $j:M\to N$ is an elementary embedding with critical point $\kappa$, then the induced $M$-ultrafilter is weakly amenable. In a striking contrast with the characterization of weakly compact cardinals, it is inconsistent to assume that *every* $\kappa$-model $M$ has a weakly amenable $M$-ultrafilter! Looking at this from the perspective of the corresponding elementary embeddings $j:M\to N$, this happens because there is too much reflection between $M$ and $N$ for objects of size $\kappa$.

The existence of weakly amenable $M$-ultrafilters for some weak $\kappa$-models characterizes Ramsey cardinals. A cardinal $\kappa$ is Ramsey whenever every $A\subseteq\kappa$ is an element of a weak $\kappa$-model $M$ which has a weakly amenable countably complete $M$-ultrafilter. If we assume that every $A\subseteq\kappa$ is an element of a $\kappa$-model $M$ which has such an $M$-ultrafilter, then we get a stronger large cardinal notion, the *strongly Ramsey* cardinal. If we further assume that every $A\subseteq\kappa$ is an element of a $\kappa$-model $M\prec H_{\kappa^+}$ for which there is such an $M$-ultrafilter, then we get an even stronger notion, the *super Ramsey* cardinal. Both notions are still weaker than a measurable cardinal. If we instead weaken our requirements and assume that every $A\subseteq\kappa$ is an element of a weak $\kappa$-model for which there is a weakly amenable $M$-ultrafilter with a well-founded ultrapower, we get a *weakly Ramsey* cardinal, which sits between ineffable and Ramsey cardinals. I introduced these notions and showed that a super Ramsey cardinal is a limit of strongly Ramsey cardinals, which is in turn a limit of Ramsey cardinals, which is in turn a limit of weakly Ramsey cardinals, which is in turn a limit of completely ineffable cardinals [2]. I also called weakly Ramsey cardinals *$1$-iterable* because they are the first step in a hierarchy of *$\alpha$-iterable* cardinals for $\alpha\leq\omega_1$, which all sit below a Ramsey cardinal (see [3] for definitions and properties). What happens if we consider intermediate versions between Ramsey and strongly Ramsey cardinals where we stratify the closure on the model $M$, considering models with $M^\alpha\subseteq M$ for cardinals $\alpha<\kappa$? What happens if we consider models $M\prec H_\theta$ for large $\theta$ and not just models $M\prec H_{\kappa^+}$?

Obviously we cannot have a weak $\kappa$-model $M$ elementary in $H_\theta$ for $\theta>\kappa^+$. So let’s drop the requirement of transitivity from the definition of a weak $\kappa$-model, but only require that $\kappa+1\subseteq M$. Now it makes sense to ask for a weak $\kappa$-model $M\prec H_\theta$ for arbitrarily large $\theta$. Suppose $\alpha\leq\kappa$ is a regular cardinal. Holy and Schlicht defined that $\kappa$ is *$\alpha$-Ramsey* if for every $A\subseteq\kappa$ and arbitrarily large regular $\theta>\kappa$, there is a weak $\kappa$-model $M\prec H_\theta$, closed under $\lt\alpha$-sequences, with $A\in M$ for which there is a weakly amenable $M$-ultrafilter on $\kappa$ (in the lone case $\alpha=\omega$, add that the ultrapower must be well-founded) [1]. It is not difficult to see that it is equivalent to require that the models exist for *all* regular $\theta>\kappa$. Also, for a fixed $\theta$, it suffices to have a single such weak $\kappa$-model $M\prec H_\theta$, meaning that the requirement that every $A$ is an element of such a model is superfluous. An $\omega$-Ramsey cardinal is a limit of weakly Ramsey cardinals, and I showed that it is weaker than a 2-iterable cardinal, and hence much weaker than a Ramsey cardinal. An $\omega_1$-Ramsey cardinal is a limit of Ramsey cardinals. A $\kappa$-Ramsey cardinal is a limit of super Ramsey cardinals. I will say where the strongly Ramsey cardinals fit in below.

It turns out that the $\alpha$-Ramsey cardinals have a game theoretic characterization! To motivate it, let’s consider the following natural strengthening of the characterization of weakly compact cardinals. Suppose that whenever $M$ is a weak $\kappa$-model, $F$ is an $M$-ultrafilter and $N$ is another weak $\kappa$-model extending $M$, then we can find an $N$-ultrafilter $\bar F\supseteq F$. What is the strength of this property? I showed that it is inconsistent. Roughly, it implies the existence of too many weakly amenable $M$-ultrafilters, which we already saw leads to inconsistency (see [1] for proof). So here is instead a game version of extending models and filters formulated by Holy and Schlicht.

Let us say that a *filter* is any subset of $P(\kappa)$ with the property that the intersection of any finite number of its elements has size $\kappa$. We will say that a filter $F$ *measures* $A\subseteq \kappa$ if $A\in F$ or $\kappa\setminus A\in F$ and we will say that $F$ *measures* $X\subseteq P(\kappa)$ if $F$ measures all $A\in X$. If $M$ is a weak $\kappa$-model, we will say that a filter $F$ is *$M$-normal* if $F\cap M$ is an $M$-ultrafilter.

Suppose $\kappa$ is weakly compact. Given an ordinal $\alpha\leq\kappa^+$ and a regular $\theta>\kappa$, consider the following two-player game of perfect information $G^\theta_\alpha(\kappa)$. Two players, the *challenger* and the *judge*, take turns to play $\subseteq$-increasing sequences $\langle M_\gamma\mid \gamma<\alpha\rangle$ of $\kappa$-models, and $\langle F_\gamma\mid\gamma<\alpha\rangle$ of filters on $\kappa$, such that the following hold for every $\gamma<\alpha$.

- The challenger plays $M_\gamma\prec H_\theta$, and then the judge plays a filter $F_\gamma$ on $\kappa$ that measures $P(\kappa)\cap M_\gamma$.
- $\langle M_{\bar\gamma}\mid \bar\gamma<\gamma\rangle,\,\langle F_{\bar \gamma}\cap M_{\bar\gamma}\mid\bar\gamma<\gamma\rangle\in M_\gamma$.

Let $M_\alpha=\bigcup_{\gamma<\alpha}M_\gamma$ and $F_\alpha=\bigcup_{\gamma<\alpha}F_\gamma$. If $F_\alpha$ is a $M_\alpha$-normal filter, then the judge wins, and otherwise the challenger wins. Note that in order to have any hope of winning the judge must play a filter $F_\gamma$ at each stage such that $F_\gamma\cap M_\gamma$ is an $M_\gamma$-ultrafilter.

Holy and Schlicht showed that if the challenger has a winning strategy in $G^\theta_\alpha(\kappa)$ for a single $\theta$, then the challenger has a winning strategy for all $\theta$, and similarly for the judge. Thus, we will say that $\kappa$ has the *$\alpha$-filter property* if the challenger has no winning strategy in the game $G^\theta_\alpha(\kappa)$ for some (all) regular $\theta>\kappa$. [1]

Holy and Schlicht showed that for regular $\alpha>\omega$, $\kappa$ has the $\alpha$-filter property if and only if $\kappa$ is $\alpha$-Ramsey! Using the game characterization, they showed that $\kappa$ is $\alpha$-Ramsey ($\alpha>\omega$) if and only if every $A\in H_{2^{\kappa^+}}$ is an element of a weak $\kappa$-model $M\prec H_{2^{\kappa^+}}$, closed under $\lt\alpha$-sequences, for which there is an $M$-ultrafilter. [1] Thus, we actually only need a single $\theta=2^{\kappa^+}$! So instead of $H_{\kappa^+}$ as in the definition of super Ramsey cardinals, the natural stopping point is $H_{2^{\kappa^+}}$. With the new characterization, we can also show that a strongly Ramsey cardinal is a limit of $\alpha$-Ramsey cardinals for every $\alpha<\kappa$.

So now we have in order of increasing strength: weakly Ramsey, $\omega$-Ramsey, $\alpha$-iterable for $2\leq\alpha\leq\omega_1$, Ramsey, $\alpha$-Ramsey for $\omega_1\leq\alpha<\kappa$, strongly Ramsey, super Ramsey, $\kappa$-Ramsey, measurable.

Why the restriction $\gamma>\omega$? I showed that an $\omega$-Ramsey cardinal is a limit of cardinals with the $\omega$-filter property (see [1] for proof). The problem arises because even if the judge wins the game $G^\theta_\omega(\kappa)$, the ultrapower of $M_\omega$ by $F_\omega$ need not be well-founded. The same problem arises for any singular cardinal of cofinality $\omega$. The solution seems to be to consider a stronger version of the game for cardinals $\alpha$ of cofinality $\omega$, where it is required that the final filter $F_\alpha$ produces a well-founded ultrapower. Let’s call this game $wfG^\theta_\alpha(\kappa)$. The well-founded games don’t seem to behave as nicely as $G^\theta_\alpha(\kappa)$. For instance, it is not known whether having a winning strategy for a single $\theta$ is equivalent to having a winning strategy for all $\theta$. I conjecture that it is not the case. Still with the well-founded games, the arguments now generalize to show that $\kappa$ is $\omega$-Ramsey if and only if $\kappa$ has the well-founded $\omega$-filter property for every $\theta$.

Finally, what about $\alpha$-Ramsey cardinals for singular $\alpha$? Well, since a weak $\kappa$-model $M$ that is closed under $\lt\alpha$-sequences for a singular $\alpha$ is also closed under $\lt\alpha^+$-sequences, $\alpha$-Ramsey for a singular $\alpha$ implies $\alpha^+$-Ramsey. So instead Holy and Schlicht defined that $\kappa$ is $\alpha$-Ramsey for a singular $\alpha$ if $\kappa$ has the well-founded $\alpha$-filter property (the well-founded part is only needed for $\alpha$ of cofinality $\omega$) [1]. Now we have the $\alpha$-Ramsey hierarchy for all cardinals $\alpha\leq\kappa$. Holy and Schlicht showed that this is a strict hierarchy of large cardinal notions: if $\kappa$ is $\alpha$-Ramsey and $\beta<\alpha$, then $V_\kappa$ is a model of proper class many $\beta$-Ramsey cardinals, and moreover if $\beta$ is regular, then $\kappa$ is indeed a limit of $\beta$-Ramsey cardinals [1].

[Bibtex]

```
@ARTICLE{HolySchlicht:HierarchyRamseyLikeCardinals,
AUTHOR= {Peter Holy and Philipp Schlicht},
TITLE= {A hierarchy of {R}amsey-like cardinals},
Note ={To appear in Fundamenta Mathematicae},
}
```

[Bibtex]

```
@ARTICLE {gitman:ramsey,
AUTHOR = {Victoria Gitman},
TITLE = {{R}amsey-like cardinals},
JOURNAL = {The Journal of Symbolic Logic},
VOLUME = {76},
YEAR = {2011},
NUMBER = {2},
PAGES = {519-540},
EPRINT={0801.4723},
PDF={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E55},
MRNUMBER = {2830415 (2012e:03110)},
MRREVIEWER = {Bernhard A. K{\"o}nig},
DOI = {10.2178/jsl/1305810762},
URL = {http://dx.doi.org/10.2178/jsl/1305810762},
}
```

[Bibtex]

```
@ARTICLE{gitman:welch,
AUTHOR= "Victoria Gitman and Philip D. Welch",
TITLE= "Ramsey-like cardinals {II}",
JOURNAL = {The Journal of Symbolic Logic},
VOLUME = {76},
YEAR = {2011},
NUMBER = {2},
PAGES = {541-560},
PDF={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf},
EPRINT ={1104.4448},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E55},
MRNUMBER = {2830435 (2012e:03111)},
MRREVIEWER = {Bernhard A. K{\"o}nig},
DOI = {10.2178/jsl/1305810763},
URL = {http://dx.doi.org/10.2178/jsl/1305810763},
}
```