TITLE: Relations between two tower numbers
QUESTION [8 upvotes]: A tower is a subset $T\subset [\omega]^\omega$ of the family $[\omega]^\omega$ of all infinite subsets of $\omega$ such that $T$ is well-ordered by the relation $\supset^*$ of almost inclusion and has no infinite pseudointersections. A tower is regular if its cardinality is a regular cardinal.
Consider two small uncountable cardinals:
$\mathfrak t=\min\{|T|:T\subset[\omega]^\omega$ is a tower$\}$;
$\hat{\mathfrak t}=\sup\{|T|:T\subset[\omega]^\omega$ is a regular tower$\}$.
It is clear that $$\mathfrak t\le\hat{\mathfrak t}\le\mathfrak c.$$
Martin's Axiom implies $\mathfrak t=\hat{\mathfrak t}=\mathfrak c$. On the other hand the strict inequality $\mathfrak t<\hat{\mathfrak t}$ is known to be consistent (but I do not know the reference).
The cardinal $\mathfrak t$, called the tower number, is well-studied in Set Theory.
I am interested in the cardinal $\hat{\mathfrak t}$, more precisely in its relation to other known cardinal characteristics of the continuum.

Problem 1. Is there any non-trivial upper or lower bound for the cardinal $\hat{\mathfrak t}$? In particular, is $\mathfrak b\le\hat{\mathfrak t}$?
Problem 2. In which known models of ZFC does the strict inequality $\mathfrak t<\hat{\mathfrak t}$ hold?

Maybe there is some fixed standard notation for $\hat{\mathfrak t}$?

REPLY [2 votes]: Assuming I understand the definitions correctly, I can give you a couple of references.
(1) Dordal (see below) gives a model in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and all towers have cardinality $\aleph_1$.  Thus, in his model $\mathfrak{t}=\hat{\mathfrak{t}}=\aleph_1<\aleph_2=\mathfrak{b}=\mathfrak{c}$, and so the last question in your first problem has a negative answer.
Edit: In fact, $\mathfrak{h}=\aleph_2$ in the model since he is using Mathias forcing, so I'm not sure of any good candidate for a non-trivial lower bound for $\hat{\mathfrak{t}}.$
(2) On the other hand, the model of Blass and Shelah discussed here gives a model in which $\mathfrak{t}=\mathfrak{b}=\aleph_1$ and there is a tower of length $\aleph_2$ (namely, the generating tower for the simple $P_{\aleph_2}$-point).  Thus, this gives a model where $\mathfrak{t}<\hat{\mathfrak{t}}$.

Edit 3/17/20
I tracked down another reference (due to Dordal [2]) with some more information on this question.  Theorem 1.3 in the paper gives a connection between towers in the structures $^\omega\omega$ and $[\omega]^\omega$:

Theorem  (Dordal, Theorem 1.3 of [2])
Let $\kappa$ be an uncountable regular cardinal, and suppose there is
  a $\kappa$-tower in $^\omega\omega$ but not $[\omega]^\omega$.  Then
  there is a $\kappa$-scale in $^\omega\omega$ (that is,
  $\mathfrak{b}=\mathfrak{d}=\kappa$).

With regard to Problem 1, we may say that if $\mathfrak{b}<\mathfrak{d}$ then $\mathfrak{b}\leq\hat{\mathfrak{t}}$ (apply the above theorem, and use the fact that there is a tower of length $\mathfrak{b}$ in the structure $^\omega\omega$. On the other hand, it is consistent that $\mathfrak{b}=\mathfrak{d}=\aleph_2$ and $\hat{\mathfrak{t}}=\aleph_1$ (this occurs in the model from [1]).
The paper has many other forcing constructions showing that not much more can be said.
[1] Dordal, Peter Lars, A model in which the base-matrix tree cannot have cofinal branches, J. Symb. Log. 52, 651-664 (1987). ZBL0637.03049.
[2] Dordal, Peter Lars, Towers in $[\omega]^{\omega}$ and $^{\omega}\omega$, Ann. Pure Appl. Logic 45, No. 3, 247-276 (1989). ZBL0686.03024.