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 $\hat{\mathfrak t}$?



Relations between two tower numbers
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](https://mathoverflow.net/questions/352034/a-question-on-simple-p-aleph-2-points/352661#352661) 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**](http://dx.doi.org/10.2307/2274354), J. Symb. Log. 52, 651-664 (1987). [ZBL0637.03049](https://zbmath.org/?q=an:0637.03049).


[2] *Dordal, Peter Lars*, [**Towers in $[\omega]^{\omega}$ and $^{\omega}\omega$**](http://dx.doi.org/10.1016/0168-0072(89)90038-9), Ann. Pure Appl. Logic 45, No. 3, 247-276 (1989). [ZBL0686.03024](https://zbmath.org/?q=an:0686.03024).


