Let us consider the following maximality principle:
$(MP_*):$ For all uncountable regular cardinals $\kappa, 2^{<\kappa}=\kappa^{+}$ and all trees of height and size $\kappa$ are specialized.
It is easily seen that the principle implies the following:
1) There are no inaccessible cardinals.
2) For all infinite cardinals $\kappa, 2^\kappa=\kappa^{++}$.
3) There are no $\kappa$-Souslin trees, for all uncountable regular cardinals $\kappa.$
Also using ideas of Todorcevic (see Some combinatorial properties of trees) one can show that the $(MP_*)$ implies the following:
4) If $P$ is a non-trivial forcing notion and if it adds a fresh subset of an uncountable regular cardinal $\kappa,$ then $P$ collapses $\kappa$ or $\kappa^+.$
Question 1. Is $(MP_*)$ consistent?
Recall that the Foreman's maximality principle says that any non-trivial forcing notion either adds a new real or collapses some cardinals.
Question 2. Does $(MP_*)$ imply the Foreman's maximality principle?
What other non-trivial consequences $(MP_*)$ has?
Remark 1. A tree $T$ of size and height $\kappa^+$ is called special if there exists $f: T \to \kappa$ such that $f(t)=f(u)=f(w)$ and $t \leq_T u, w$ implies $u \leq_T w$ or $w \leq_T u.$
Remark 2. By results of Todorcevic and independently Baumgartner, ($MP_*$) is consistent for $\kappa=\aleph_1.$ On the other hand, shelah and I have proved its consistency for successor a regular cardinal.
