**Update:** Perhaps the question is too difficult. I would appreciate, thus, even just comments or related observations.

This question assumes familiarity with combinatorial cardinal characteristics of the continnum. It is the essence of an earlier question.

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 $\kappa$ be a cardinal number.
A *tower of height $\kappa$* is a $\kappa$-sequence
$\langle\, s_\alpha : \alpha<\kappa\,\rangle$ in $[\mathbb{N}]^\infty$ such that

- This $\kappa$-sequence is $\subset^*$-decreasing as the ordinal number $\alpha$ increases.
- The set $\{\,s_\alpha : \alpha<\kappa\,\}$ has no pseudointersection. (That is, there is no infinite set $s$ such that $s\subseteq^* s_\alpha$ for all $\alpha<\kappa$).

An element $a\in [\mathbb{N}]^\infty$ is identified with its increasing enumeration. This way, the set $[\mathbb{N}]^\infty$
becomes the family of increasing functions in $\mathbb{N}^\mathbb{N}$,
and the standard relation $\le^*$ is defined on
$[\mathbb{N}]^\infty$.
A set $X\subseteq [\mathbb{N}]^\infty$ is *bounded* if it is bounded (from above) with respect to $\le^*$.

The general goal is to understand when is there an *unbounded* tower of height $\mathfrak{b}$. Let us call this axiom BT.

It is known or easy to see that:

- An unbounded set has no pseudointersection. So we may remove the need for no pseudointersection from the definition of
*tower*without altering BT. - If there is an unbounded tower of any cardinality, then BT holds.
- If $\mathfrak{t}=\mathfrak{b}$ or $\mathfrak{b}<\mathfrak{d}$, then BT holds.

**Open-ended question.** Can the axiom BT be expressed using (standard)
cardinal characteristics of the continuum?

Ashutosh proved that BT is consistent with "$\aleph_1=\mathfrak{t}<\mathfrak{b}=\mathfrak{c}=\aleph_2$".

Will Brian points out that that BT fails in the Hechler model. BT also fails in the Laver model, indirectly by the main result of the linked paper. I suspect that BT also fails in the Mathias model.

**Question 1.** Does any additional inequality or inequality among cardinals of the continuum (one not following from
$\mathfrak{t}=\mathfrak{b}$ or $\mathfrak{b}<\mathfrak{d}$) imply BT?

**Question 2.** Does BT imply any equality among cardinals of the continuum?

Since BT follows from CH, the hypotheisis BT does not imply any *in*equality.

**Motivation.** BT implies that, even in the realm of real sets, the selective covering property $\operatorname{S}_1(\Gamma,\Gamma)$ (which is consistently trivial) is nontrivial.