# The Special Aronszajn Tree Property at $\kappa^+$ and $\square_{\kappa, 2}$

## TL;DR

This paper proves the consistency of the square principle 2 with all 2-Aronszajn trees being special for any regular kappa, extending results to all regular cardinals using advanced set-theoretic methods.

## Contribution

It establishes the consistency of 2 with the special Aronszajn tree property at 22 for all regular cardinals, a significant extension in set theory.

## Key findings

- 2 is consistent with all 2-Aronszajn trees being special.
- The result applies simultaneously to all regular 2.
- It shows 2 cannot generally be strengthened to 2.

## Abstract

We show that for any regular cardinal $\kappa$, $\square_{\kappa, 2}$ is consistent with "all $\kappa^+$-Aronszajn trees are special." By a result of Shelah and Stanley this is optimal in the sense that $\square_{\kappa, 2}$ may not be strengthened to $\square_{\kappa}$. Using methods of Golshani and Hayut we obtain our consistency result simultaneously for all regular $\kappa$.