This talk discusses ongoing joint work with my thesis supervisor Boban Velickovic.
We first isolate a particular family of countable elementary submodels of $H_\theta$ which are characterised using certain open games by the demand that the second player has a winning strategy. We will then analyse the existence of (projective stationary) many such models and discuss the connection with precipitousness of the nonstationary ideal on $\omega_1$. Next, we discuss a particular forcing $\mathbb{P}$ that consists of finite conditions in which these special models feature as side conditions. We survey some interesting properties that this forcing shares with an $\mathcal{L}-forcing$ defined by Claverie-Schindler and a Namba-like forcing defined by Ketchersid-Larson-Zapletal, to both of which it shows resemblance. It will follow that this side condition approach using special models gives yet another way to increase the second uniform indiscernible in a stationary set preserving way beyond some arbitrary prespecified ordinal.