Games and Ramsey-like cardinals

Dan Saattrup Nielsen, Philip Welch

Published 2018-04-27Version 1

We generalise the $\alpha$-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals $\alpha$ to arbitrary ordinals, 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 $\omega$-Ramseys are also strategic $\omega$-Ramsey, and that strategic $\omega_1$-Ramsey cardinals are equiconsistent with measurable cardinals, both by showing that they are measurable in $K$ and that they carry precipitous ideals. We also show that the $n$-Ramseys satisfy indescribability properties and use them to characterise ineffable-type 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).