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

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 $\kappa$.