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We show that for any regular cardinal $\kappa$, $\square_{\kappa, 2}$ isconsistent with "all $\kappa^+$-Aronszajn trees are special." By a result ofShelah and Stanley this is optimal in the sense that $\square_{\kappa, 2}$ maynot be strengthened to $\square_{\kappa}$. Using methods of Golshani and Hayutwe obtain our consistency result simultaneously for all regular $\kappa$.

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