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(This question assumes familiarity with combinatorial cardinal characteristics of the continnum.)

Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$, partially ordered by $\subseteq^*$, where $a\subseteq^* b$ means $a\setminus b$ is finite.

Let $\mathfrak{ht}$ (sometimes called $\operatorname{Depth}^+([\mathbb{N}]^\infty)$) be the minimal cardinal number $\kappa$ such that there is no $\subset^*$-decreasing $\kappa$-sequence in $[\mathbb{N}]^\infty$. Then $\mathfrak{t}<\mathfrak{ht}\le\mathfrak{c}^+$.

A classic result from Kunen's thesis asserts that, when adding $\kappa\ge\aleph_1$ Cohen reals to a model of CH, we obtain $\mathfrak{ht}=\aleph_2$.

Open-ended question. Can the hypothesis $\mathfrak{b}<\mathfrak{ht}$ be expressed using cardinal characteristics other than $\mathfrak{ht}$?

It is easy to see that "$\mathfrak{t}=\mathfrak{b}$ or $\mathfrak{b}<\mathfrak{d}$" implies "$\mathfrak{b}<\mathfrak{ht}$".

Question 1. Is it consistent that "$\aleph_1=\mathfrak{t}<\mathfrak{b}=\mathfrak{c}=\aleph_2<\mathfrak{ht}$"?

In the Laver model, we have $\aleph_1=\mathfrak{t}<\mathfrak{ht}=\mathfrak{b}=\mathfrak{c}=\aleph_2$.

Question 2. What is the value of $\mathfrak{ht}$ in the Hechler and Mathias models?

Update: Will Brian answers Question 1 in the positive below, in the Hechler model (thus also answering Question 2 for Hechler). I thought that $\mathfrak{b}<\mathfrak{ht}$ implies there is a nontrivial set of reals satisfying the selection principle $\operatorname{S}_1(\Gamma,\Gamma)$, by the the linked paper. But Brian's comments make it clear that I oversimplified the question for this purpose. I opened a new question that fits better the intended application.

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    $\begingroup$ A model for Q1 with arbitrarily large continuum can be obtained by adding $\aleph_1$ random reals to a model of MA plus not CH, $\endgroup$ – Ashutosh May 10 '16 at 15:48
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In the Hechler model, $\aleph_1 = \mathfrak{t} < \mathfrak{b} = \mathfrak{c} = \aleph_2 < \mathfrak{ht}$.

(By "the Hechler model" I mean the result of a length-$\omega_2$ finite support iteration of the forcing to adjoin a dominating real.)

That $\aleph_1 = \mathfrak{t} < \mathfrak{b} = \mathfrak{c} = \aleph_2$ is discussed in Section 11.6 of Blass's handbook article. That $\mathfrak{ht} > \mathfrak{c}$ follows from Theorem 4.1 in

J. Baumgartner and P. Dordal, "Adjoining dominating functions," Journal of Symbolic Logic vol. 50 (1985), pp. 94 - 101, available here.

Interestingly, Baumgartner and Dordal show that, while there are $\subset^*$-decreasing sequences of length $\omega_2$ in this model, none of them are maximal -- all maximal sequences have cofinality $\omega_1$.

This answers question 1 and the first part of question 2. I don't know what $\mathfrak{ht}$ is in the Mathias model, and I also don't know the answer to your open-ended question (though I think it's an interesting one).

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  • $\begingroup$ I'm missing something: Take a nonmaximal decreasing sequence of length $\mathfrak{c}$. Then there is a pseudointersection, right? So, on top of this pseudointersection you can build a tower of height $\mathfrak{t}$, and end up with a tower of height continuum. $\endgroup$ – Boaz Tsaban May 10 '16 at 17:04
  • $\begingroup$ @BoazTsaban: Yes, that construction would work. The resulting tower would have cardinality $\mathfrak{c}$, but its cofinality would still have to be $\omega_1$ (your maximal $\subset^*$-descreasing sequence might have order type $\omega_2+\omega_1$, for example). $\endgroup$ – Will Brian May 10 '16 at 17:23
  • $\begingroup$ @BoazTsaban: Perhaps it's confusing for me to use "$\mathfrak{c}$" when referring to an order type, not a cardinality. I'll edit accordingly. $\endgroup$ – Will Brian May 10 '16 at 17:26

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