Increasing the second uniform indiscernible by strongly ssp forcing

We use side condition techniques to introduce a natural stationary set preserving forcing P c c ( λ , μ ) \mathbb {P}^{\mathbf {cc}}(\lambda ,\mu ) that (under N S ω 1 \mathsf {NS}_{\omega _1} precipitous + existence of H θ # H_{\theta }^\# for a sufficiently large regular θ \theta ) increases the second uniform indiscernible u 2 \mathbf {u}_2 beyond some given ordinal λ \lambda . The forcing P c c \mathbb {P}^{\mathbf {cc}} shares this property with forcings defined by Claverie and Schindler [J. Symbolic Logic 74 (2009), pp. 187–200] and Ketchersid, Larson, and Zapletal [J. Symbolic Logic 72 (2007), pp. 1372–1378]. We show that P c c \mathbb {P}^{\mathbf {cc}} is in addition strongly ssp. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games G M c p ( X ) \mathbf {G}^{\mathbf {cp}}_M(X)