On certain indestructibility of strong cardinals and a question of Hajnal (Q1114682)

In this paper the authors solve a question of Hajnal. Starting with a model with n strong cardinals they construct a model satisfying the following: for some cardinal \(\lambda\), \(\lambda <\lambda^{\omega}<\lambda^{\omega_ 1}<...<\lambda^{\omega_ n}\) and \(2^{\omega_ i}=\omega_{i+1}\) for \(i\in \omega\). In their model, strongness of \(\kappa\) is indestructible under \(\kappa^+\)-weakly closed forcing notions satisfying the Prikry condition. On the other hand it is known that \(\{\lambda^{\delta}|\) \(2^{\delta}<\lambda \}\) is always finite.