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

Abstract.We generalise the -Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals to arbitrary ordinals , and answer several questions posed in that paper. In particular, we show that -Ramseys are downwards absolute to the core model for all $\alpha$ of uncountable cofinality, that strategic -Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic -Ramsey cardinals are equiconsistent with measurable cardinals. We also show that the -Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing connections between the -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 which are critical points of an elementary embedding , where and are -models of size which contains as both a subset and an element (we call such models *weak -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 , we can find as above such that and that the -measure induced by is furthermore *countably complete*, meaning that intersections of a countable sequences (in ) of elements of are nonempty, and *weakly amenable*, meaning for every having -cardinality .

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 -Ramsey cardinals for being a cardinal. Their definition is based upon the following game of length .

Here all the ‘s are weak -models and the ‘s are -measures. We require that the models are all elementary in some large , that the models and measures are -increasing and that, letting be the union of all the measures and the union of all the models, is required to be an -normal -measure with a well-founded ultrapower.

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