# Paper: Games and Ramsey-like cardinals

D. S. Nielsen and P. Welch, Games and Ramsey-like cardinals, 2018, manuscript under review — arXiv.

Abstract. We generalise the $\alpha$-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals $\alpha$ to arbitrary ordinals $\alpha$, and answer several questions posed in that paper. In particular, we show that $\alpha$-Ramseys are downwards absolute to the core model $K$ for all $\alpha$ of uncountable cofinality, that strategic $\omega$-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic $\omega_1$-Ramsey cardinals are equiconsistent with measurable cardinals. We also show that the $n$-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing connections between the $\alpha$-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990) and Sharpe and Welch (2011).

The Ramsey-like cardinals can loosely be described as cardinals $\kappa$ which are critical points of an elementary embedding $j:(M,\in)\to (N,\in)$, where $M$ and $N$ are $\textsf{ZFC}^-$-models of size $\kappa$ which contains $\kappa$ as both a subset and an element (we call such models weak $\kappa$-models). We can therefore see these cardinals as “mini-versions” of measurable cardinals, and indeed, a result of Kunen shows that Ramsey cardinals fall into this category as well. In this case, given any $A\subseteq\kappa$, we can find $j:M\to N$ as above such that $A\in M$ and that the $M$-measure $\mu$ induced by $j$ is furthermore countably complete, meaning that intersections of a countable sequences (in $V$) of elements of $\mu$ are nonempty, and weakly amenable, meaning $x\cap\mu\in M$ for every $x\in M$ having $M$-cardinality $\kappa$.

If we remove these conditions of countable completeness and weak amenability then we arrive at an equivalent characterisation of weakly compact cardinals, and Gitman (2011) introduced strengthenings as well called strongly Ramsey cardinals and super Ramsey cardinals. Holy and Schlicht (2018) further added onto the catalogue by introducing a whole hierarchy of $\lambda$-Ramsey cardinals for $\lambda$ being a cardinal. Their definition is based upon the following game of length $\lambda$.

Here all the $\mathcal M_\alpha$‘s are weak $\kappa$-models and the $\mu_\alpha$‘s are $\mathcal M_\alpha$-measures. We require that the models are all elementary in some large $H_\theta$, that the models and measures are $\subseteq$-increasing and that, letting $\mu$ be the union of all the measures and $\mathcal M$ the union of all the models, $\mu$ is required to be an $\mathcal M$-normal $\mathcal M$-measure with a well-founded ultrapower.

We then say that a cardinal $\kappa$ is $\lambda$-Ramsey if player I doesn’t have a winning strategy in the above game (the game doesn’t depend upon $\theta$). So this is a kind of “filter extension property”. We can also define the ostensibly stronger notion of being strategic $\lambda$-Ramsey if player II does have a winning strategy.

References

• Feng, Q. (1990). A hierarchy of Ramsey cardinals. Annals of Pure and Applied Logic, 49(2):257–277.
• Gitman, V. (2011). Ramsey-like cardinals. The Journal of Symbolic Logic, 76(2):519–540.
• Holy, P. and Schlicht, P. (2018). A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae. DOI: 10.4064/fm396-9-2017.
• Sharpe, I. and Welch, P. D. (2011). Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Annals of Pure and Applied Logic, 162:863–902.