Selection Principles in Mathematics — Problem 49

Definitions and mathematical statement

Under the conventional interpretation of the source notation, write $[{\mathbb{N}}]^{\aleph_0}$ for the family of infinite subsets of ${\mathbb{N}}$, and write $A\subseteq^*B$ when $A\setminus B$ is finite. A family $\mathcal F\subseteq[{\mathbb{N}}]^{\aleph_0}$ has the strong finite intersection property if every nonempty finite subfamily has infinite intersection. An infinite pseudointersection of $\mathcal F$ is a set $A\in[{\mathbb{N}}]^{\aleph_0}$ such that $A\subseteq^*F$ for every $F\in\mathcal F$. The pseudointersection number $\mathfrak p$ is the least cardinality of a family with the strong finite intersection property and no infinite pseudointersection. The tower number $\mathfrak t$ is the least cardinal $\kappa$ for which there is a family $(T_\alpha)_{\alpha<\kappa}$ of infinite subsets of ${\mathbb{N}}$ such that $T_\beta\subseteq^*T_\alpha$ whenever $\alpha<\beta<\kappa$, and the family has no infinite pseudointersection. These are conventional definitions; the supplied excerpt does not itself define these symbols.

The source also uses cardinal invariants $\mathfrak p_\kappa$, but the supplied record gives no definition of them. It reports the identities \[ \mathfrak p=\mathfrak p_2=\mathfrak p_3=\cdots=\mathfrak p_{\aleph_0}, \qquad \mathfrak p_{\mathfrak t}=\mathfrak t, \] and attributes $\mathfrak p=\mathfrak p_{\aleph_1}$ to the reference label [Pyt]. It displays the hierarchy \[ \mathfrak p=\mathfrak p_{\aleph_1}\leq\mathfrak p_{\aleph_2}\leq\cdots\leq\mathfrak p_{\mathfrak t}=\mathfrak t. \] These are supplied background assertions, not independently verified conclusions of this draft.

The question in Problem 49 is whether \[ \mathfrak p_{\mathfrak p}=\mathfrak t. \] Here $\mathfrak p_{\mathfrak p}$ denotes the source's indexed invariant evaluated at the cardinal $\mathfrak p$. A fully specified mathematical statement requires the missing definition of $\mathfrak p_\kappa$, including its admissible indices and defining quantifiers. The reported identities and inequalities do not supply that definition. The underlying axiomatic assumptions are also not stated in the supplied excerpt.

Short English statement

Tsaban's Problem 49 asks whether the source's cardinal invariant $\mathfrak p_\kappa$, evaluated at $\kappa=\mathfrak p$, equals the tower number $\mathfrak t$. The supplied record labels the problem open, but omits the definition of $\mathfrak p_\kappa$, so the formulation remains incomplete and requires human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID 800620, exact source problem code catalogue entry, title "Selection Principles in Mathematics --- Problem 49." The supplied record attributes the problem to Tsaban (2006) and labels its status "open"; this label is supplied metadata, not an independently verified current assessment.
• Tsaban, Selection Principles and Special Sets of Reals: Open Problems (2006), Problem 49. Supplied source URL: https://arxiv.org/abs/math/0609601. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of access or verification for this draft.
• The supplied excerpt cites [Pyt] for $\mathfrak p=\mathfrak p_{\aleph_1}$, but supplies no bibliographic details identifying that reference.