Between Ramsey and measurable cardinals

TL;DR

This paper explores the relationships between various large cardinal notions, especially Ramsey and measurable cardinals, providing new characterizations and clarifying their position in the hierarchy.

Contribution

It introduces and characterizes intermediate large cardinal notions related to measurability using filter games and hierarchy analysis.

Findings

Located baby versions of measurability within the hierarchy.

Provided characterizations of these notions via filter games.

Determined the theory of the universe up to a measurable cardinal.

Abstract

We study several intertwined hierarchies between $\kappa$-Ramsey cardinals and measurable cardinals to illuminate the structure of the large cardinal hierarchy in this region. In particular, we study baby versions of measurability introduced by Bovykin and McKenzie and some variants by locating these notions in the large cardinal hierarchy and providing characterisations via filter games. As an application, we determine the theory of the universe up to a measurable cardinal.