Forcing with Adequate Sets of Models as Side Conditions

We present a general framework for forcing on $\omega_2$ with finite conditions using countable models as side conditions. This framework is based on a method of comparing countable models as being membership related up to a large initial segment. We give several examples of this type of forcing, including adding a function on $\omega_2$, adding a nonreflecting stationary subset of $\omega_2 \cap \textrm{cof}(\omega)$, and adding an $\omega_1$-Kurepa tree

Coherent Adequate Sets and Forcing Square

We introduce the idea of a coherent adequate set of models, which can be used as side conditions in forcing. As an application we define a forcing poset which adds a square sequence on $\omega_2$ using finite conditions

Quotients of Strongly Proper Forcings and Guessing Models

We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have simple universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the $\omega_1$-approximation property. We prove that the existence of stationarily many $\omega_1$-guessing models in $P_{\omega_2}(H(\theta))$, for sufficiently large cardinals $\theta$, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss