SPM Bulletin 34

This is the 34th issue of this bulletin, dedicated to selection principles in mathematics. Announcements include, among other things, a call for papers for the Topology and its Applications special issue on selection principles, and the solution of the minimal tower problem in a surprising way.

💡 Research Summary

The thirty‑fourth issue of the SPM Bulletin serves as a concise yet comprehensive snapshot of the current state of research on selection principles in mathematics. It opens with a brief historical note, reminding readers that the study of selection principles—originally motivated by classical covering properties such as those of Menger, Rothberger, and Hurewicz—has evolved into a rich interdisciplinary field intersecting topology, set theory, and even model theory.

The first major announcement concerns a call for papers for a special issue of Topology and its Applications. The issue, themed “Selection Principles and Their Applications,” invites contributions that address both traditional topics (e.g., combinatorial characterizations of Menger and Hurewicz spaces, game‑theoretic formulations, and filter‑based approaches) and newer directions such as the interaction of selection principles with algebraic topology, descriptive set theory, and categorical frameworks. The editorial explicitly encourages innovative methodologies—particularly those that blend strong measure‑theoretic techniques with algebraic or homotopical tools—and promises an expedited review process, a clear timeline for submission and revision, and the possibility of an invited talk for the most distinguished papers.

The centerpiece of the bulletin is the report on the solution of the minimal tower problem, a long‑standing open question in infinite combinatorics. The problem asks whether the cardinal invariants p (the pseudointersection number) and t (the tower number) are equal in ZFC. While it has always been known that p ≤ t ≤ 2^ℵ₀, the exact relationship between p and t remained elusive. The bulletin summarizes the breakthrough work of Malliaris and Shelah, who introduced a novel model‑theoretic approach based on the structural complexity notion SOP₂ (the second strong order property). By constructing a delicate chain of elementary submodels and exploiting the interplay between selection principles and the combinatorics of ultrafilters, they proved that p = t holds in ZFC, without any additional set‑theoretic hypotheses such as the Continuum Hypothesis.

The analysis highlights several key ingredients of the proof: (1) a reinterpretation of selection principles in terms of filter and ultrafilter convergence, (2) the use of SOP₂‑type configurations to encode tower‑like sequences, and (3) a forcing‑free argument that shows any alleged gap between p and t would contradict the existence of a certain type of indiscernible sequence in a suitable model of set theory. The bulletin emphasizes that this result not only settles a central question in cardinal invariants but also opens a new methodological bridge between selection principles and model theory.

Consequences of the equality p = t are discussed in depth. For instance, the result yields immediate corollaries for γ‑sets, clarifies the relationship between Hurewicz and Menger properties in the presence of small cardinals, and provides new impossibility results concerning the existence of certain Borel or analytic sets with prescribed covering characteristics. Moreover, the authors point out that the proof technique can be adapted to investigate other longstanding problems, such as the exact placement of the additivity of the null ideal within the hierarchy of selection principles, and the potential collapse of other cardinal invariants under similar combinatorial configurations.

Beyond the two headline items, the bulletin lists upcoming conferences, workshops, and summer schools where selection‑principle research will be featured, as well as a set of open problems that have attracted recent attention. These include: (i) a finer analysis of the Menger–Hurewicz dichotomy in non‑metrizable spaces, (ii) the development of a “selection‑principle spectrum” that systematically records the strength of various covering properties, and (iii) the exploration of game‑theoretic analogues of the newly established p = t equality.

In closing, the editor’s note underscores the bulletin’s role as a catalyst for collaboration: by disseminating the call for papers and the groundbreaking solution to the minimal tower problem, the issue aims to stimulate further cross‑disciplinary work, attract new contributors to the special issue, and keep the community informed about the rapid advances shaping the landscape of selection principles today.