Games and Ramsey-like 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).