Games and Ramsey-like cardinals

TL;DR

This paper extends the concept of $eta$-Ramsey cardinals to all ordinals, explores their properties, and establishes their equiconsistencies with other large cardinals, providing new insights into their structure and relationships.

Contribution

It generalizes $eta$-Ramsey cardinals to all ordinals, proves their downward absoluteness, and links them to other large cardinal notions through equiconsistency results and game-theoretic characterizations.

Findings

$eta$-Ramsey cardinals are downward absolute to $K$ for uncountable cofinality.

Strategic $ extomega$-Ramsey cardinals are equiconsistent with remarkable cardinals.

Strategic $eta$-Ramsey cardinals are equiconsistent with measurable cardinals.

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 $\alpha$-Ramsey cardinals are equiconsistent with measurable cardinals for all $\alpha>\omega$. 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 further connections between the $\alpha$-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990) and Sharpe and Welch (2011).