Increasing the second uniform indiscernible by strongly ssp forcing

We use side condition techniques to introduce a natural stationary set preserving forcing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P Superscript bold c bold c Baseline left-parenthesis lamda comma mu right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">c</mml:mi> <mml:mi mathvariant="bold">c</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> λ </mml:mi> <mml:mo>,</mml:mo> <mml:mi> μ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {P}^{\mathbf {cc}}(\lambda ,\mu )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that (under <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper N sans-serif upper S Subscript omega 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">N</mml:mi> <mml:mi mathvariant="sans-serif">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathsf {NS}_{\omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> precipitous + existence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript theta Superscript number-sign"> <mml:semantics> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> θ </mml:mi> </mml:mrow> <mml:mi mathvariant="normal"> # </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">H_{\theta }^\#</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a sufficiently large regular <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi> θ </mml:mi> <mml:annotation encoding="application/x-tex">\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) increases the second uniform indiscernible <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold u 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">u</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbf {u}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> beyond some given ordinal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi> λ </mml:mi> <mml:annotation encoding="application/x-tex">\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The forcing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P Superscript bold c bold c"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">c</mml:mi> <mml:mi mathvariant="bold">c</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {P}^{\mathbf {cc}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P Superscript bold c bold c"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">c</mml:mi> <mml:mi mathvariant="bold">c</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {P}^{\mathbf {cc}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is in addition <italic>strongly</italic> ssp. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper G Subscript upper M Superscript bold c bold p Baseline left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">G</mml:mi> </mml:mrow> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">c</mml:mi> <mml:mi mathvariant="bold">p</mml:mi> </mml:mrow> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {G}^{\mathbf {cp}}_M(X)</mml