The tree property at all regular even cardinals

TL;DR

This paper constructs models of ZFC set theory where the tree property holds at all regular even cardinals, using large cardinal assumptions, thus answering several open questions in set theory.

Contribution

It introduces new models demonstrating the tree property at all regular even cardinals under specific large cardinal assumptions, expanding understanding of combinatorial properties in set theory.

Findings

Tree property holds at all regular even cardinals in the constructed models.

Models are built assuming a strong cardinal and measurable cardinals above it.

Answers to open questions by Friedman, Honzik, Stejskalova, and others are provided.

Abstract

Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of $ZFC$ in which for every singular cardinal $\delta$, $\delta$ is strong limit, $2^\delta=\delta^{+3}$ and the tree property at $\delta^{++}$ holds. This answers a question of Friedman, Honzik and Stejskalova [8]. We also produce, relative to the existence of a strong cardinal and two measurable cardinals above it, a model of $ZFC$ in which the tree property holds at all regular even cardinals. The result answers questions of Friedman-Halilovic [5] and Friedman-Honzik [6].