INDESTRUCTIBILITY OF THE TREE PROPERTY

In the first part of the article, we show that if $\omega \le \kappa < \lambda$ are cardinals, ${\kappa ^{ < \kappa }} = \kappa$ , and λ is weakly compact, then in $V\left[M {\left( {\kappa ,\lambda } \right)} \right]$ the tree property at $$\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $$ is indestructible under all ${\kappa ^ + }$ -cc forcing notions which live in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$ , where ${\rm{Add}}\left( {\kappa ,\lambda } \right)$ is the Cohen forcing for adding λ -many subsets of κ and $\left( {\kappa ,\lambda } \right)$ is the standard Mitchell forcing for obtaining the tree property at $\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $ . This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that λ is supercompact and generalize the construction and obtain a model ${V^{\rm{*}}}$ , a generic extension of V , in which the tree property at ${\left( {{\kappa ^{ + + }}} \right)^{{V^{\rm{*}}}$ is indestructible under all ${\kappa ^ + }$ -cc forcing notions living in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$ , and in addition under all forcing notions living in ${V^{\rm{*}}}$ which are ${\kappa ^ + }$ -closed and “liftable” in a prescribed sense (such as ${\kappa ^{ + + }}$ -directed closed forcings or well-met forcings which are ${\kappa ^{ + + }}$ -closed with the greatest lower bounds).