arXiv subjects
Publications and source records attributed to Curial Gallart.
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 $\mathrm{PFA}$.
We force the existence of a chain of length $\omega_3$ in $[\omega_1]^{\omega_1}$ increasing modulo finite. The construction involves symmetric systems of models of two types as side conditions, introduced by the second author. This improves previous results of Koszmider and Veli\v{c}kovi\'{c}-Venturi.
The purpose of this paper is to present a general method for forcing on $\omega_2$ and $\omega_3$ with finite conditions, while preserving all cardinals and some fragments of $\mathrm{GCH}$. This method is based on the technique of forcing with finite symmetric systems of elementary submodels, and improves earlier versions of this forcing by including models of two types. We will present several applications of the pure side condition forcing and variants thereof, by adding a Kurepa tree on $\omega_2$, a club subset of $\omega_2$ that avoids infinite sets from the ground model, a function bounding every canonical function below $\omega_3$ on a club, and a simplified $(\omega_2,1)$-morass.