The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $\kappa $ is strongly compact if and only if the strong tree property holds at $\kappa $, and supercompact if and only if ITP holds at $\kappa $. We present several results motivated by the problem of obtaining the strong tree property and ITP at many successive cardinals simultaneously; these results focus on the successors of singular cardinals. We describe a general class of forcings that will obtain the strong tree property and ITP at the successor of a singular cardinal of any cofinality. Generalizing a result of Neeman about the tree property, we show that it is consistent for ITP to hold at $\aleph _n$ for all $2 \leq n < \omega $ simultaneously with the strong tree property at $\aleph _{\omega +1}$; we also show that it is consistent for ITP to hold at $\aleph _n$ for all $3 < n < \omega $ and at $\aleph _{\omega +1}$ simultaneously. Finally, turning our attention to singular cardinals of uncountable cofinality, we show that it is consistent for the strong and super tree properties to hold at successors of singulars of multiple cofinalities simultaneously.

We present an alternative proof that from large cardinals, we can force the tree property at $\kappa ^+$ and $\kappa ^{++}$ simultaneously for a singular strong limit cardinal $\kappa $ . The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $\kappa =\aleph _{\omega ^2}$ .

We study the notion of tightly stationary sets which was introduced by Foreman and Magidor in [8]. We obtain two consistency results showing that certain sequences of regular cardinals ${\langle {\kappa _n}\rangle _{n < \omega }}$ can have the property that in some generic extension, every ground-model sequence of fixed-cofinality stationary sets ${S_n} \subseteq {\kappa _n}$ is tightly stationary. The results are obtained using variations of the short-extenders forcing method.