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 ) Pᶜᶜ(λ ,μ ) that (under N S ω 1 NSω _1 precipitous + existence of H θ # Hθ^# for a sufficiently large regular θ θ ) increases the second uniform indiscernible u 2 u_2 beyond some given ordinal λ λ . The forcing P c c Pᶜᶜ 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 Pᶜᶜ 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 ) Gᶜᵖ_M(X)</mml