Games and Ramsey-like cardinals
Dan Saattrup Nielsen, Philip Welch

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 -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 of uncountable cofinality, that strategic -Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic -Ramsey cardinals are equiconsistent with measurable cardinals for all . 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 further connections between the -Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990) and Sharpe and Welch (2011).
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