Abstract
In the first part of the article, we show that if omega = <= kappa < lambda are cardinals, kappa(<kappa) = kappa, and lambda is weakly compact, then in V[M(kappa, lambda)] the tree property at lambda = (kappa(++))(V[M(kappa,lambda)]) is indestructible under all kappa(+)-cc forcing notions which live in V[Add(kappa, lambda)], where Add(kappa, lambda) is the Cohen forcing for adding lambda-many subsets of kappa and M(kappa, lambda) is the standard Mitchell forcing for obtaining the tree property at lambda = (kappa(++))(V[M(kappa, lambda)]). This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants.
In the second part, we assume that lambda is supercompact and generalize the construction and obtain a model V*, a generic extension of V, in which the tree property at (kappa(++))(V)* is indestructible under all kappa(+)-cc forcing notions living in V[Add(kappa, lambda)], and in addition under all forcing notions living in V* 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).