Does $\mathfrak{p}(\kappa)=\mathfrak{t}(\kappa)$ hold for every uncountable regular cardinal $\kappa$? (the higher analogue of Malliaris–Shelah's $\mathfrak{p}=\mathfrak{t}$)

The pseudointersection number $\mathfrak{p}$ and the tower number $\mathfrak{t}$ are classical cardinal characteristics of the continuum, and their equality — proved at $\omega$ by Malliaris and Shelah in 2016 — is one of the deepest structural theorems of the subject. This problem is its direct higher analogue. For an uncountable regular cardinal $\kappa$, define $\mathfrak{p}(\kappa)$ as the least size of a family of $\kappa$-sized subsets of $\kappa$ with the strong intersection property (every subfamily of size $<\kappa$ has intersection of size $\kappa$) but no $\kappa$-sized pseudointersection, and $\mathfrak{t}(\kappa)$ as the least size of such a family that is moreover well-ordered by $\subseteq^*$ (a tower). It is known that $\kappa^+\le\mathfrak{p}(\kappa)\le\mathfrak{t}(\kappa)\le 2^\kappa$, that — assuming $\kappa^{<\kappa}=\kappa$ — $\mathfrak{p}(\kappa)=\mathfrak{t}(\kappa)$ whenever $\mathfrak{p}(\kappa)=\kappa^+$ or $\operatorname{cf}(2^\kappa)\in\{\kappa^+,\kappa^{++}\}$, and that, under the same hypothesis, either $\mathfrak{p}(\kappa)=\mathfrak{t}(\kappa)$ or there exists a $(\mathfrak{p}(\kappa),\lambda)$-gap of club-supported slaloms for some $\lambda<\mathfrak{p}(\kappa)$; yet neither a ZFC proof of equality in general nor a model with a strict inequality is known. Posed by Schilhan (2019) and re-posed by Fischer–Montoya–Schilhan–Soukup (2022, Question 5.1), the problem is open already for $\kappa=\aleph_1$ and for strongly inaccessible $\kappa$ with $\kappa^{<\kappa}=\kappa$.

Background

Cardinal characteristics of the continuum measure how small a family of subsets of the natural numbers can be while exhibiting a given combinatorial behaviour. Two of the most central are the pseudointersection number $\mathfrak{p}$ and the tower number $\mathfrak{t}$. Write $A\subseteq^*B$ to mean that $A\setminus B$ is finite. A family $\mathcal{F}\subseteq[\omega]^\omega$ has the strong finite intersection property if any finitely many of its members have infinite intersection; an infinite set $S$ is a pseudointersection of $\mathcal{F}$ if $S\subseteq^*F$ for every $F\in\mathcal{F}$. Then $\mathfrak{p}$ is the least size of such a family with no infinite pseudointersection, and $\mathfrak{t}$ is the least length of a $\subseteq^*$-decreasing well-ordered sequence (a tower) with no infinite pseudointersection. Classically $\mathfrak{p}\le\mathfrak{t}$, and after several decades in which a strict inequality seemed plausible, Malliaris and Shelah proved in 2016, through their cofinality spectrum theorem from model theory — a result constraining the possible cofinalities of ultraproducts of finite structures — that $\mathfrak{p}=\mathfrak{t}$ — a landmark result for which they received the Hausdorff medal.

In parallel, set theorists have developed the combinatorics of the generalized Baire spaces: for an uncountable regular cardinal $\kappa$, the function space $\kappa^\kappa$ and Cantor space $2^\kappa$ carry the bounded ($<\kappa$-box) topology, in which basic opens fix an initial segment of length $<\kappa$; the ideal $\mathcal{M}_\kappa$ of $\kappa$-meager sets consists of unions of $\kappa$ many nowhere dense sets, and its additivity $\operatorname{add}(\mathcal{M}_\kappa)$ — the least number of $\kappa$-meager sets whose union is not $\kappa$-meager — and covering number $\operatorname{cov}(\mathcal{M}_\kappa)$ — the least number of $\kappa$-meager sets needed to cover the space — generalize the classical meager-ideal invariants, alongside the higher bounding number $\mathfrak{b}(\kappa)$ and dominating number $\mathfrak{d}(\kappa)$ defined via eventual domination $f\le^*g$ (meaning $|\{\alpha<\kappa:g(\alpha)<f(\alpha)\}|<\kappa$). The naive transfer of $\mathfrak{p}$ and $\mathfrak{t}$ to $\kappa$ collapses: at uncountable $\kappa$ there are countable $\subseteq^*$-decreasing families of $\kappa$-sized sets with no $\kappa$-sized pseudointersection (partition $\kappa$ into $\kappa$-sized pieces $X_n$, $n<\omega$, and take tails $\bigcup_{m\ge n}X_m$), so without extra hypotheses the higher tower number would be trivially $\aleph_0$ (the collapse of the SIP-free variants $\mathfrak{p}^*,\mathfrak{t}^*$ is worked out by Brian and Verner, as recorded by Fischer–Montoya–Schilhan–Soukup). The standard cure, adopted by Garti, by Schilhan — the source of this problem — and by all subsequent work, is to require the strong intersection property at $\kappa$ (SIP): every subfamily of size $<\kappa$ must have intersection of size $\kappa$. Under this definition a tower is a $\subseteq^*$-well-ordered SIP family with no $\kappa$-sized pseudointersection, and one always has $\kappa^+\le\mathfrak{p}(\kappa)\le\mathfrak{t}(\kappa)\le 2^\kappa$.

Several partial results delimit the question. Shelah and Spasojević (2002), who began the study of the higher invariants $\mathfrak{b}_\kappa,\mathfrak{t}_\kappa$, proved a 'pity on $\lambda$' theorem: if $\kappa\le\mu<\mathfrak{t}(\kappa)$ then $2^\mu=2^\kappa$; they also produced the first model with $\mathfrak{p}(\kappa)<\mathfrak{b}(\kappa)$. Garti (2011) proved, assuming $\kappa^{<\kappa}=\kappa$: if $\mathfrak{p}(\kappa)=\kappa^+$ then $\mathfrak{t}(\kappa)=\kappa^+$; if $\operatorname{cf}(2^\kappa)\in\{\kappa^+,\kappa^{++}\}$ then $\mathfrak{p}(\kappa)=\mathfrak{t}(\kappa)$; and $\operatorname{cf}(\mathfrak{p}(\kappa))\ne\kappa$. Schilhan (2019) proved $\mathfrak{t}(\kappa)\le\mathfrak{b}(\kappa)$ (in fact $\mathfrak{t}(\kappa)\le\operatorname{add}(\mathcal{M}_\kappa)$ when $\kappa^{<\kappa}=\kappa$, and equality of the club-filter variants $\mathfrak{p}_{cl}(\kappa)=\mathfrak{t}_{cl}(\kappa)=\mathfrak{b}(\kappa)$, where these are the analogues with the family restricted to club — closed unbounded — subsets of $\kappa$), and proved a higher analogue of Bell's theorem — the classical result that any $\sigma$-centered partial order (a countable union of centered pieces) has a filter meeting every family of fewer than $\mathfrak{p}$ dense sets — making $\mathfrak{p}(\kappa)$ the density threshold of the corresponding forcing axiom for $\kappa$-specially centered partial orders (a strengthening of $\kappa$-centeredness, the analogue of $\sigma$-centeredness with $\kappa$ directed pieces). Fischer–Montoya–Schilhan–Soukup (2022) showed that $\mathfrak{p}(\kappa)$ is provably regular, proved the dichotomy that, when $\kappa^{<\kappa}=\kappa$, either $\mathfrak{p}(\kappa)=\mathfrak{t}(\kappa)$ or a $(\mathfrak{p}(\kappa),\lambda)$-gap of club-supported slaloms exists for some $\lambda<\mathfrak{p}(\kappa)$ (a slalom is a function assigning to each $\alpha<\kappa$ a small subset of $\alpha$; the existence of such gaps is itself open), and built a new model of $\mathfrak{p}(\kappa)=\kappa^+<\mathfrak{p}_{cl}(\kappa)=2^\kappa$. The surrounding infrastructure is actively mapped: Brendle–Brooke-Taylor–Friedman–Montoya (2018) developed the Cichoń diagram — the classical web of ZFC-provable inequalities among the meager-related invariants — for $\mathcal{M}_\kappa$ at strongly inaccessible $\kappa$ (documenting that the classical preservation theorems do not lift, since limit stages of cofinality $<\kappa$ break the usual inductive forcing arguments), Brooke-Taylor–Fischer–Friedman–Montoya (2017) and Honzík–Stejskalová (2025) produced models with nontrivial values of the higher invariants at supercompact and inaccessible $\kappa$ (in particular $\kappa^+<\mathfrak{t}(\kappa)=\mathfrak{u}(\kappa)<2^\kappa$ is consistent, where $\mathfrak{u}(\kappa)$ is the least size of a base for a $\kappa$-complete ultrafilter on $\kappa$), and Benhamou–Goldberg (2026) calibrated the consistency strength of tower-type statements at measurable $\kappa$. Ben-Neria–Garti (2020) state explicitly that it is unknown whether a strict inequality $\mathfrak{p}(\kappa)<\mathfrak{t}(\kappa)$ is consistent for any $\kappa$. What is missing, in either direction, is exactly the content of the problem below.

Problem Statement

Let $\kappa$ be an uncountable regular cardinal. For $A,B\subseteq\kappa$ write $A\subseteq^*B$ iff $|A\setminus B|<\kappa$, and write $[\kappa]^\kappa$ for the family of subsets of $\kappa$ of cardinality $\kappa$. A family $\mathcal{F}\subseteq[\kappa]^\kappa$ has the strong intersection property (SIP) if $|\bigcap\mathcal{F}'|=\kappa$ for every $\mathcal{F}'\subseteq\mathcal{F}$ with $|\mathcal{F}'|<\kappa$. A set $S\in[\kappa]^\kappa$ is a pseudointersection of $\mathcal{F}$ if $S\subseteq^*F$ for every $F\in\mathcal{F}$. Define $\mathfrak{p}(\kappa)=\min\{|\mathcal{F}|:\mathcal{F}\subseteq[\kappa]^\kappa\text{ has the SIP and no pseudointersection}\}$ and $\mathfrak{t}(\kappa)=\min\{|\mathcal{T}|:\mathcal{T}\subseteq[\kappa]^\kappa\text{ is }\subseteq^*\text{-well-ordered, has the SIP, and has no pseudointersection}\}.$ Both minima exist and satisfy $\kappa^+\le\mathfrak{p}(\kappa)\le\mathfrak{t}(\kappa)\le 2^\kappa$.

Question (Schilhan 2019; Fischer–Montoya–Schilhan–Soukup 2022, Question 5.1): Is it a theorem of ZFC that $\mathfrak{p}(\kappa)=\mathfrak{t}(\kappa)$ for every uncountable regular cardinal $\kappa$? Equivalently, since $\mathfrak{p}(\kappa)\le\mathfrak{t}(\kappa)$ is provable: is there a model of ZFC in which $\mathfrak{p}(\kappa)<\mathfrak{t}(\kappa)$ for some uncountable regular $\kappa$? The question is open already for $\kappa=\aleph_1$ and for strongly inaccessible $\kappa$ with $\kappa^{<\kappa}=\kappa$, the regime in which the club filter (the filter of closed unbounded subsets of $\kappa$) and the $\kappa$-meager ideal $\mathcal{M}_\kappa$ (the ideal of unions of $\kappa$ many nowhere dense subsets of $\kappa^\kappa$) behave most like their countable-level counterparts. Partial restrictions are known — equality holds whenever $\mathfrak{p}(\kappa)=\kappa^+$ or $\operatorname{cf}(2^\kappa)\in\{\kappa^+,\kappa^{++}\}$ (Garti, assuming $\kappa^{<\kappa}=\kappa$), and in every model constructed to date in which both invariants have been computed — but the general ZFC status is unresolved in both directions.

References

1. Jonathan Schilhan, "Generalised pseudointersections", Mathematical Logic Quarterly 65(4):479–489, 2019. DOI: 10.1002/malq.201800084, https://doi.org/10.1002/malq.201800084 (source paper)
2. Maryanthe Malliaris and Saharon Shelah, "Cofinality spectrum theorems in model theory, set theory, and general topology", Journal of the American Mathematical Society 29(1):237–297, 2016. DOI: 10.1090/jams830, https://doi.org/10.1090/jams830
3. Saharon Shelah and Zoran Spasojević, "Cardinal invariants $\mathfrak{b}_\kappa$ and $\mathfrak{t}_\kappa$", Publications de l'Institut Mathématique (Beograd) 72(86):1–9, 2002. DOI: 10.2298/PIM0272001S, https://doi.org/10.2298/PIM0272001S
4. Shimon Garti, "Pity on $\lambda$", preprint, 2011. arXiv:1103.1947, https://arxiv.org/abs/1103.1947
5. Omer Ben-Neria and Shimon Garti, "On configurations concerning cardinal characteristics at regular cardinals", Journal of Symbolic Logic 85(2):691–708, 2020. DOI: 10.1017/jsl.2019.80, https://doi.org/10.1017/jsl.2019.80 (arXiv:1905.06067, https://arxiv.org/abs/1905.06067)
6. Vera Fischer, Diana Carolina Montoya, Jonathan Schilhan, and Dániel T. Soukup, "Towers and gaps at uncountable cardinals", Fundamenta Mathematicae 257(2):141–166, 2022. DOI: 10.4064/fm109-9-2021, https://doi.org/10.4064/fm109-9-2021 (arXiv:1906.00843, https://arxiv.org/abs/1906.00843)
7. Jörg Brendle, Andrew Brooke-Taylor, Sy-David Friedman, and Diana Carolina Montoya, "Cichoń's diagram for uncountable cardinals", Israel Journal of Mathematics 225(2):959–1010, 2018. DOI: 10.1007/s11856-018-1688-y, https://doi.org/10.1007/s11856-018-1688-y (arXiv:1611.08140, https://arxiv.org/abs/1611.08140)
8. Andrew D. Brooke-Taylor, Vera Fischer, Sy-David Friedman, and Diana C. Montoya, "Cardinal characteristics at $\kappa$ in a small $\mathfrak{u}(\kappa)$ model", Annals of Pure and Applied Logic 168(1):37–49, 2017. DOI: 10.1016/j.apal.2016.08.004, https://doi.org/10.1016/j.apal.2016.08.004 (arXiv:1511.02687, https://arxiv.org/abs/1511.02687)
9. Vera Fischer and Silvan Horvath, "Generalized tower spectra", Journal of Symbolic Logic, 2025, 1–17. DOI: 10.1017/jsl.2024.61, https://doi.org/10.1017/jsl.2024.61
10. Radek Honzík and Šárka Stejskalová, "Generalized cardinal invariants for an inaccessible $\kappa$ with compactness at $\kappa^{++}$", Archive for Mathematical Logic 64(7–8):1077–1102, 2025. DOI: 10.1007/s00153-025-00977-2, https://doi.org/10.1007/s00153-025-00977-2 (arXiv:2308.13478, https://arxiv.org/abs/2308.13478)
11. Tom Benhamou and Gabriel Goldberg, "Measures that violate the generalized continuum hypothesis", Advances in Mathematics 487:110778, 2026. DOI: 10.1016/j.aim.2025.110778, https://doi.org/10.1016/j.aim.2025.110778 (arXiv:2503.20094, https://arxiv.org/abs/2503.20094)
12. Ömer Faruk Bağ and Vera Fischer, "Non-linear iterations and higher splitting", preprint, 2020. arXiv:2005.11105, https://arxiv.org/abs/2005.11105