Is the auxiliary cardinal $\mathfrak{d}^*$ equal to $\mathfrak{d}$? Optimal tower-refinement of filter-bases after $\mathfrak{p}=\mathfrak{t}$

Hirschorn's preprint A Measure Theoretic Proof of $\mathfrak{p}=\mathfrak{t}$ (arXiv:2212.14171) introduces the auxiliary cardinal $\mathfrak{d}^*$: the least size of a family $[\mathcal{H}]$ in the Boolean algebra $\mathcal{P}(\mathbb{N})/\mathrm{Fin}$ whose coideal $\mathrm{co}([\mathcal{H}])$ — the elements meeting every member of $[\mathcal{H}]$, the "thick" sets — contains a filter of character below $\mathfrak{p}$ with no lower bound in $\mathrm{co}([\mathcal{H}])$. The preprint proves $\mathfrak{p}\le\mathfrak{b}\le\mathfrak{d}^*\le\mathfrak{d}$ and thereby answers Rothberger's 1948 question affirmatively: if $\mathfrak{b}>\mathfrak{p}$, every filter-base of cardinality $\mathfrak{p}$ can be refined to a tower. What remains open, and what this problem records, are the optimal bounds on $\mathfrak{d}^*$: (i) is $\mathfrak{d}^*=\mathfrak{d}$ in ZFC (Question 3 of the preprint; proved there only under $\mathfrak{p}=\aleph_1$); (ii) does $\mathfrak{p}<\mathfrak{d}$ alone — rather than $\mathfrak{p}<\mathfrak{b}$ or $\mathfrak{p}<\mathfrak{d}^*$ — already guarantee that every filter-base of cardinality $\mathfrak{p}$ can be refined to a tower (Question 4, equivalently: does $\mathfrak{p}<\mathfrak{d}$ imply $\mathfrak{p}<\mathfrak{lr}$, where $\mathfrak{lr}$ is Tsaban's linear refinement number); and (iii) Tsaban's question whether $\mathfrak{lr}\le\mathfrak{d}^*$ (Question 5). A positive answer to (i) settles (ii) and (iii); a negative answer to (iii) refutes (i). All three questions remain open.

Background

Write $A\subseteq^*B$ if $A\setminus B$ is finite. A family $\mathcal{F}$ of infinite subsets of $\mathbb{N}$ is a filter-base (equivalently, centered) if every finite subfamily has infinite intersection, and an infinite set $x$ is a pseudo-intersection of $\mathcal{F}$ if $x\subseteq^*a$ for all $a\in\mathcal{F}$. The pseudo-intersection number $\mathfrak{p}$ is the least size of a filter-base with no pseudo-intersection. A tower is a family $\{a_\xi:\xi<\theta\}$ well-ordered by $\supseteq^*$ (i.e. $a_\xi\supseteq^*a_\eta$ whenever $\xi<\eta$) with no infinite pseudo-intersection, and $\mathfrak{t}$ is the least length $\theta$ of a tower; $\mathfrak{p}\le\mathfrak{t}$ because a tower is itself a filter-base. On $\mathbb{N}^{\mathbb{N}}$ put $f\le^*g$ if $f(n)\le g(n)$ for all but finitely many $n$; $\mathfrak{b}$ is the least size of an unbounded family in $(\mathbb{N}^{\mathbb{N}},\le^*)$ and $\mathfrak{d}$ the least size of a dominating (cofinal) family. Rothberger (1948) proved $\mathfrak{t}\le\mathfrak{b}$, that $\mathfrak{p}=\aleph_1$ implies $\mathfrak{p}=\mathfrak{t}$, and that if $\mathfrak{p}=\aleph_1$ and $\mathfrak{b}>\mathfrak{p}$ then every filter-base of cardinality $\mathfrak{p}$ can be refined to a tower — a refinement of a family $\mathcal{B}$ being a family $\mathcal{A}$ such that every $b\in\mathcal{B}$ has an $a\in\mathcal{A}$ with $a\subseteq^*b$ — and he asked whether the latter holds without the hypothesis $\mathfrak{p}=\aleph_1$: if $\mathfrak{b}>\mathfrak{p}$, can every filter-base of cardinality $\mathfrak{p}$ be refined to a tower? Malliaris and Shelah (2016) proved $\mathfrak{p}=\mathfrak{t}$ in ZFC using model-theoretic cofinality spectrum theorems. A positive answer to Rothberger's refinement question is strictly stronger than the equality $\mathfrak{p}=\mathfrak{t}$: refining a filter-base of size $\mathfrak{p}$ (one with no pseudo-intersection, which exists by the definition of $\mathfrak{p}$) to a tower of length at most $\mathfrak{p}$ forces $\mathfrak{t}\le\mathfrak{p}$, hence $\mathfrak{p}=\mathfrak{t}$.

Tsaban captured the refinement phenomenon with the linear refinement number $\mathfrak{lr}$: the least cardinality of a filter-base with no $\subseteq^*$-linearly-ordered refinement. Then $\mathfrak{p}\le\mathfrak{lr}\le\mathfrak{d}$, and the statement "every filter-base of cardinality $\mathfrak{p}$ can be refined to a tower" is equivalent to $\mathfrak{p}<\mathfrak{lr}$: at size $\mathfrak{lr}$ the witnessing family has no linear, hence no tower, refinement, while every filter-base of size below $\mathfrak{lr}$ does have a tower refinement — one with no pseudo-intersection is refined by a tower of height $\mathfrak{p}$ by a lemma of Machura–Shelah–Tsaban stated just below, and one with a pseudo-intersection $y$ is refined by any tower of subsets of $y$, which exists because $\mathfrak{t}=\mathfrak{p}$ (every subset of $y$ is almost contained in every member of the family). Machura, Shelah and Tsaban proved that $\mathfrak{lr}=\mathfrak{d}$ in every model whose continuum is at most $\aleph_2$, that $\mathfrak{lr}=\aleph_1$ implies $\mathfrak{d}=\aleph_1$, that $\mathfrak{p}<\mathfrak{b}$ implies $\mathfrak{lr}\le\mathfrak{b}$, and that every centered family of size below $\mathfrak{lr}$ either has a pseudo-intersection or is refined by a tower of height exactly $\mathfrak{p}$; by forcing, $\mathfrak{lr}$ is not provably equal to $\mathfrak{d}$ and is independent of many classical characteristics (consistently $\mathfrak{lr}<\mathfrak{d}$, and consistently $\mathfrak{p}<\mathfrak{lr}$, for instance $\mathfrak{p}<\mathfrak{lr}<\mathfrak{q}_0$ where $\mathfrak{q}_0$ is the least size of a non-$Q$-set of reals).

Hirschorn's preprint answers Rothberger's question affirmatively through a new cardinal. For $G$ contained in a Boolean algebra, $\mathrm{co}(G)$ denotes the set of elements meeting every member of $G$ — the coideal of the filter that $G$ generates; relative to a family $\mathcal{H}$ of infinite subsets of $\mathbb{N}$, the members of $\mathrm{co}([\mathcal{H}])$ are called thick sets. The character of a filter is the least size of a filter-base generating it. The preprint defines $\mathfrak{d}^*$ as the least cardinality of a family $[\mathcal{H}]\subseteq\mathcal{P}(\mathbb{N})/\mathrm{Fin}$ such that $\mathrm{co}([\mathcal{H}])$ contains a filter of character less than $\mathfrak{p}$ with no lower bound in $\mathrm{co}([\mathcal{H}])$; equivalently, the least $|\mathcal{H}|$ for which there is a filter-base, of size below $\mathfrak{p}$, consisting of thick sets but having no thick pseudo-intersection. (The character restriction is essential: without it the single set $[\mathbb{N}]$ would witness size 1, since $\mathcal{P}(\mathbb{N})/\mathrm{Fin}$ has a filter of character $\mathfrak{p}$ with no nonzero lower bound by the very definition of $\mathfrak{p}$.) The preprint establishes $\mathfrak{p}\le\mathfrak{d}^*$, that $\mathfrak{d}^*\le\mathfrak{d}$ with equality when $\mathfrak{p}=\aleph_1$, and that $\mathfrak{p}<\mathfrak{d}^*$ implies every filter-base of cardinality $\mathfrak{p}$ can be refined to a tower (in contrapositive form: $\mathfrak{p}=\mathfrak{lr}$ implies $\mathfrak{p}=\mathfrak{d}^*$). Its main theorem is $\mathfrak{b}\le\mathfrak{d}^*$, proved via "imaginary" measures that exist only in forcing extensions collapsing $\aleph_1$; combined with $\mathfrak{t}\le\mathfrak{b}$ this yields $\mathfrak{p}=\mathfrak{t}$ and the affirmative answer to Rothberger's question. The preprint stops short of deciding whether $\mathfrak{d}^*=\mathfrak{d}$, and that gap is the problem recorded here.

Problem Statement

Determine the optimal bounds on $\mathfrak{d}^*$ — its exact position relative to $\mathfrak{d}$ and to $\mathfrak{lr}$ in ZFC. Concretely, answer Questions 3, 4 and 5 of Hirschorn's preprint:

(1) Is $\mathfrak{d}^*=\mathfrak{d}$ in ZFC? The equality is known when $\mathfrak{p}=\aleph_1$; the question is whether it holds without any hypothesis on $\mathfrak{p}$.

(2) If $\mathfrak{p}<\mathfrak{d}$, can every filter-base of cardinality $\mathfrak{p}$ be refined to a tower? Equivalently: does $\mathfrak{p}<\mathfrak{d}$ imply $\mathfrak{p}<\mathfrak{lr}$? Failure would be exactly the consistency of $\mathfrak{p}=\mathfrak{lr}<\mathfrak{d}$.

(3) Is $\mathfrak{lr}\le\mathfrak{d}^*$? (Tsaban's question.)

Logical relations among the three, using the preprint's results: a positive answer to (1) implies positive answers to (2), since $\mathfrak{p}<\mathfrak{d}=\mathfrak{d}^*$ and $\mathfrak{p}<\mathfrak{d}^*$ suffices for tower refinement by the preprint's Lemma 8, and to (3), since Tsaban proved $\mathfrak{lr}\le\mathfrak{d}$; a negative answer to (3) implies a negative answer to (1), as it would give $\mathfrak{d}^*<\mathfrak{lr}\le\mathfrak{d}$; and a negative answer to (2), i.e. a model of $\mathfrak{p}=\mathfrak{lr}<\mathfrak{d}$, yields $\mathfrak{p}=\mathfrak{d}^*<\mathfrak{d}$ by the contrapositive of Lemma 8. The single scientific objective is to locate $\mathfrak{d}^*$, that is, to replace the sufficient condition $\mathfrak{p}<\mathfrak{d}^*$ for tower refinement by the optimal one.

References

1. James Hirschorn, "A Measure Theoretic Proof of $\mathfrak{p}=\mathfrak{t}$", preprint, arXiv:2212.14171 (v1 December 2022, v3 May 2023). DOI: 10.48550/arXiv.2212.14171; https://arxiv.org/abs/2212.14171.
2. Michał Machura, Saharon Shelah, Boaz Tsaban, "The linear refinement number and selection theory", Fundamenta Mathematicae 234 (2016), 15–40. DOI: 10.4064/fm124-8-2015; arXiv:1404.2239; https://arxiv.org/abs/1404.2239.
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8. Claudio Agostini, Andrea Medini, Lyubomyr Zdomskyy, "Countable spaces, realcompactness, and the pseudointersection number", Journal of Symbolic Logic (published online 2024). DOI: 10.1017/jsl.2024.52; arXiv:2310.17984; https://arxiv.org/abs/2310.17984.
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