(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](http://dx.doi.org/10.1090/S0002-9939-10-10407-9) 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](http://dx.doi.org/10.1090/S0002-9939-10-10407-9) 
$\aleph\_1=\mathfrak{t}<\mathfrak{ht}=\mathfrak{b}=\mathfrak{c}=\aleph\_2$.


**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](http://dx.doi.org/10.1090/S0002-9939-10-10407-9). But Brian's comments make it clear that I oversimplified the question for this purpose. I opened a [new question](https://mathoverflow.net/questions/238500/when-does-the-unbounding-number-mathfrakb-embed-in-mathbbn-infty) that fits better the intended application. 



What is the height (or depth) of $[\mathbb{N}]^\infty$?
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](http://www.math.lsa.umich.edu/~ablass/hbk.pdf). That $\mathfrak{ht} > \mathfrak{c}$ follows from Theorem 4.1 in



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> > J. Baumgartner and P. Dordal, "Adjoining dominating functions," *Journal of Symbolic Logic* vol. **50** (1985), pp. 94 - 101, available [here](http://www.jstor.org/stable/2273792?seq=1#page_scan_tab_contents).
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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).


