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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.

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    $\begingroup$ I am confused. If $\kappa=\tau^+$, then the condition in $(MP_*)$ gives $2^\tau=\tau^{+3}$. $\endgroup$ – Péter Komjáth Apr 10 '17 at 5:28
  • $\begingroup$ You are right, I edited it. $\endgroup$ – Mohammad Golshani Apr 10 '17 at 5:35
  • $\begingroup$ If we view $\kappa$ itself as a linear order, then it is a tree of height and size $\kappa$, but it cannot be special. Doesn't this show that $\text{MP}_*$ is inconsistent? $\endgroup$ – Joel David Hamkins Apr 10 '17 at 11:34
  • $\begingroup$ No, I don't take such objects as trees, I may also say that by special tree I mean something a little different. See Todorcevic's paper mentioned above in which the consistency of the statement for $\aleph_1$ is stated. For higher successors of regulars, Shelah and I have proved the consistency of the statement. $\endgroup$ – Mohammad Golshani Apr 10 '17 at 11:53
  • $\begingroup$ See statement (C) in his paper. $\endgroup$ – Mohammad Golshani Apr 10 '17 at 11:54

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