Increasing the second uniform indiscernible by strongly ssp forcing

We introduce a new and natural stationary set preserving forcing \(\mathbb P^{c-c}({\lambda},{\mu})\) that (under \(\mathsf{NS}_{\omega_1}\) precipitous + existence of \(H_{\theta}^#\) for a sufficiently large regular \({\theta}\)) increases the second uniform indiscernible \(\mathbf{u}_2\) beyond some given ordinal \({\lambda}\). The forcing \(\mathbb P^{c-c}\) shares this property with forcings defined in [2] and [9]. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games \(\mathbf{G}_M^{cap}(X)\) and the catching-capturing games \(\mathbf{G}_M^{c-c}(X)\). In particular, these games are used to isolate a special family of countable elementary submodels \(M \prec H_{\theta}\) that occur as side conditions in \(\mathbb P^{c-c}\) and thus allow to control the forcing in a strong way.