In the Handbook of Set Theory, Foreman has (essentially) the following proposition (3.9):
Suppose $\kappa_n>...>\kappa_0$ and $\lambda_n>...>\lambda_0$ are regular cardinals and $(\kappa_n,...,\kappa_0) \twoheadrightarrow (\lambda_n,...,\lambda_0)$. If either GCH holds or there are countably many cardinals between $\lambda_0$ and $\kappa_n$, then for all structures $\frak A$ on $\kappa_n$, there is $\frak B \prec \frak A$ such that $| \frak B \cap \kappa_i | = \lambda_i$ for all $i \leq n$, and $\lambda_0 \subseteq \frak B$.
Question 1: Can we drop the assumptions of GCH or countably many cardinals between the top and bottom cardinal?
Question 2: The argument I know using GCH also uses the regularity of the $\kappa_i$'s. But it is superfluous in the case of countably many cardinals between. So can we drop the regularity assumption in the GCH case?
Question 3: The stronger Chang conjecture can be shown indestructible under $\lambda_0^+$-c.c. forcing. Does $\lambda_0^+$-c.c. forcing preserve the ordinary Chang conjecture? What about forcing of size $\lambda_0$?
