Strong failures of club guessing at the successor of a regular cardinal

Abstract

Starting from a model of $\mathrm{ZFC}$, we force the simultaneous failure of the Very Weak Club Guessing and the $\mho$ principles at $S_{\kappa }^{{\kappa }^{+}}$, for any regular cardinal $\kappa$. Our results are obtained by means of a forcing iteration technique, due to Krueger, that incorporates models as side conditions. At ${\omega }_1$, the failure of these principles is a well-known consequence of $\mathrm{PFA}$.

32 pages