# A note on the eightfold way

## TL;DR

This paper constructs a set-theoretic model assuming a Mahlo cardinal where certain combinatorial properties of $ ext{cof}( ext{omega})$ subsets of $ ext{omega}_2$ are demonstrated, including the existence of an $ ext{omega}_2$-Aronszajn tree.

## Contribution

It introduces a new model under large cardinal assumptions where specific reflection and approachability properties fail or hold, advancing understanding of set-theoretic combinatorics.

## Key findings

- Existence of an $ ext{omega}_2$-Aronszajn tree in the model
- Failure of the $ ext{omega}_1$-approachability property
- Stationary subsets of $ ext{omega}_2 ext{∩ cof}( ext{omega})$ reflect

## Abstract

Assuming the existence of a Mahlo cardinal, we construct a model in which there exists an $\omega_2$-Aronszajn tree, the $\omega_1$-approachability property fails, and every stationary subset of $\omega_2 \cap \mathrm{cof}(\omega)$ reflects.