Given a cardinal \kappa we consider the statement "\kappa is threadable + \square_\kappa fails". Assuming "CH + \kappa=\omega_2" or "2^\omega=\omega_2 + \kappa=\omega_3", a core model induction argument then yields "AD_R + \Theta is regular". This brings the results by Jensen-Schindler-Schimmerling-Steel one cardinal down. This is a joint work with Nam Trang.