A hierarchy of Ramsey-like cardinals

TL;DR

This paper develops a hierarchy of large cardinals situated between weakly compact and measurable cardinals, using infinite filter games and elementary embeddings, to better understand Ramsey-like cardinals and their relations.

Contribution

It introduces a new hierarchy of large cardinals related to Ramsey-like cardinals, with multiple equivalent characterizations, expanding the understanding of their position in set theory.

Findings

Hierarchy of large cardinals established

Equivalent characterizations via filter games and embeddings

Positions of Ramsey-like cardinals clarified

Abstract

We introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the Ramsey-like cardinals introduced by Victoria Gitman, and is based on certain infinite filter games, however also has a range of equivalent characterizations in terms of elementary embeddings. The aim of this paper is to locate the Ramsey-like cardinals studied by Gitman, and other well-known large cardinal notions, in this hierarchy.