Pseudointersection and tower numbers of topological Ramsey spaces: the Dobrinen–Navarro Flores questions on $\mathfrak{p}_\mathcal{R}\le\mathfrak{p}$, $\mathfrak{p}_\mathcal{R}=\mathfrak{t}_\mathcal{R}$, and values outside $\{\omega_1,\mathfrak{p}\}$

For a $\sigma$-closed topological Ramsey space $(\mathcal{R},\le,\le^*,r)$ one defines a space-specific pseudointersection number $\mathfrak{p}_\mathcal{R}$ and tower number $\mathfrak{t}_\mathcal{R}$ by transplanting the classical definitions of $\mathfrak{p}$ and $\mathfrak{t}$ from the almost-inclusion order on $[\omega]^\omega$ to the natural $\sigma$-closed coarsening $\le^*$ of the Ramsey space order. In every case computed to date one has $\mathfrak{p}_\mathcal{R}=\mathfrak{t}_\mathcal{R}\in\{\omega_1,\mathfrak{p}\}$. Dobrinen and Navarro Flores posed three questions that remain open: whether $\mathfrak{p}_\mathcal{R}\le\mathfrak{p}$ holds for every topological Ramsey space; whether some space has $\mathfrak{p}_\mathcal{R}\ne\mathfrak{t}_\mathcal{R}$; and whether some space has $\mathfrak{p}_\mathcal{R}\notin\{\omega_1,\mathfrak{p}\}$. No resolution of these questions is known in the literature to date. The problem asks whether the Malliaris–Shelah phenomenon $\mathfrak{p}=\mathfrak{t}$ and the bound $\mathfrak{p}_\mathcal{R}\le\mathfrak{p}$ are inherent to the class of $\sigma$-closed topological Ramsey spaces, or whether a Ramsey space can separate them.

Background

A family $\mathcal{F}\subseteq[\omega]^\omega$ of infinite subsets of the natural numbers has the strong finite intersection property (SFIP) if every finite subfamily has infinite intersection; a pseudointersection of $\mathcal{F}$ is an infinite set $Y$ almost contained in every member, $Y\subseteq^* X$ for all $X\in\mathcal{F}$, where $Y\subseteq^* X$ means that $Y\setminus X$ is finite. The pseudointersection number $\mathfrak{p}$ is the least cardinality of an SFIP family with no pseudointersection, and the tower number $\mathfrak{t}$ is the least length of a $\subseteq^*$-decreasing chain (a tower) with no pseudointersection. These are classical cardinal characteristics of the continuum satisfying $\omega_1\le\mathfrak{p}\le\mathfrak{t}\le 2^{\aleph_0}$, and the equality $\mathfrak{p}=\mathfrak{t}$, a longstanding open problem, was proved by Malliaris and Shelah via their cofinality spectrum theorems.

A topological Ramsey space is a partially ordered set of infinite sequences of finite approximations, written $(\mathcal{R},\le,r)$ with $r_n(X)$ the finite initial segment of $X$ of length $n$, satisfying four structural axioms (A.1–A.4) isolated by Todorcevic; the axioms guarantee an abstract form of Ellentuck's theorem, namely that every subset of $\mathcal{R}$ with the Baire property in the associated Ellentuck-type topology is Ramsey, i.e. for every $X\in\mathcal{R}$ there is $Y\le X$ whose entire basic neighborhood is either contained in or disjoint from the set. Standard examples include the Ellentuck space on $[\omega]^\omega$, the spaces $\mathrm{FIN}_k^{[\infty]}$ of infinite block sequences of $k$-blocks — finite-support functions $f:\omega\to\{0,\dots,k\}$ whose range contains $k$, arranged with pairwise disjoint, successively increasing supports — the spaces generated by Fraïssé classes of finite structures (classes of finite structures closed under isomorphism, substructure, and amalgamation, whose countable homogeneous limit carries the induced ordering), and the Carlson–Simpson space of variable words, in which a condition is a finite word over an alphabet enlarged by variable symbols and is refined by substituting variable-free words for its variables.

The forcing connection is the point of departure for the source paper. These spaces carry a natural coarsening $\le^*$ of $\le$, the abstract analogue of $\subseteq^*$, and Dobrinen and Navarro Flores call $(\mathcal{R},\le,\le^*,r)$ a $\sigma$-closed topological Ramsey space when $\le^*$ is $\sigma$-closed (every countable descending chain of conditions has a lower bound in $\mathcal{R}$) and the two orders have isomorphic separative quotients: the separative quotient of a partial order is obtained by identifying two conditions each compatible with the other, so two orders with isomorphic separative quotients give rise to the same forcing. Forcing with $(\mathcal{R},\le^*)$ is then a $\sigma$-closed forcing that adds ultrafilters with finite Ramsey degree: the Ramsey degree $t(\mathcal{U},n)$ of an ultrafilter $\mathcal{U}$ on $\omega$ is the least $t$ such that every finite coloring of the $n$-element subsets of $\omega$ takes at most $t$ colors on $[X]^n$ for some $X\in\mathcal{U}$. Computing these degrees for forcings such as $\mathcal{P}(\omega^k)/\mathrm{Fin}^{\otimes k}$ is the first half of the paper.

Because $(\mathcal{R},\le^*)$ is $\sigma$-closed, the classical definitions of $\mathfrak{p}$ and $\mathfrak{t}$ transplant verbatim (Definitions 23–24 of the source): $\mathfrak{p}_\mathcal{R}$ is the least cardinality of a subfamily $\mathcal{F}\subseteq\mathcal{R}$ with the $\le^*$-SFIP (every finite subfamily has a common $\le^*$-lower bound in $\mathcal{R}$) but with no $Y\in\mathcal{R}$ satisfying $Y\le^* X$ for all $X\in\mathcal{F}$, and $\mathfrak{t}_\mathcal{R}$ is the least length of a $\le^*$-tower with no such pseudointersection. $\sigma$-closedness gives $\omega_1\le\mathfrak{p}_\mathcal{R}\le\mathfrak{t}_\mathcal{R}$, and the source paper proves a dichotomy for the known examples. For spaces with enough internal independence one gets the value $\mathfrak{p}$. The Independent Extension Property (IEP) — roughly, that any finite approximation to a condition can be further extended using coordinates of any prescribed member of the space — yields the lower bound $\mathfrak{p}\le\mathfrak{p}_\mathcal{R}$ (Theorem 14), via Bell's theorem identifying $\mathfrak{p}$ with the forcing number $\mathfrak{m}(\sigma\text{-centered})$ for $\sigma$-centered partial orders (a partial order is $\sigma$-centered if it is a countable union of suborders in which any finitely many conditions have a common lower bound). The stronger Independent Sequences of Structures property (ISS*), which requires that the finite structure underlying each finite approximation embed into every later one, gives the full equality $\mathfrak{p}_\mathcal{R}=\mathfrak{t}_\mathcal{R}=\mathfrak{p}$ (Theorem 16), the upper bound coming from transferring classical towers and SFIP families between $([\omega]^\omega,\subseteq^*)$ and the space and applying Malliaris–Shelah. This covers the spaces $\mathcal{R}_\alpha$ dense in Laflamme's transfinite hierarchy of $\sigma$-closed ultrafilter forcings ($1\le\alpha<\omega_1$), the spaces generated by Fraïssé classes with the Ramsey property, and — by a separate argument (Theorem 15) — the spaces $\mathrm{FIN}_k^{[\infty]}$ for all finite $k$. In contrast, for the high- and infinite-dimensional Ellentuck spaces $E_\alpha$, $2\le\alpha<\omega_1$ (forcing-equivalent to the quotient Boolean algebras $\mathcal{P}(\omega^\alpha)/\mathrm{Fin}^{\otimes\alpha}$, where $\mathrm{Fin}^{\otimes\alpha}$ is the iterated tensor product of the ideal of finite sets, defined recursively so that a set is small when all but finitely many of its vertical sections are small in the previous ideal), for the spaces $E_B$ associated with uniform barriers (a barrier is a family $B$ of finite subsets of $\omega$ such that every infinite subset of $\omega$ has exactly one initial segment in $B$; it is uniform when all its members have the same length), and for the Carlson–Simpson space, one gets $\mathfrak{p}_\mathcal{R}=\mathfrak{t}_\mathcal{R}=\omega_1$, building on older computations of Szymański–Zhou and Matet and independently obtained for finite dimensions by Kurilić. Thus in every case computed so far the two invariants are equal and take one of the two extreme values $\omega_1$ or $\mathfrak{p}$, and the proofs of the $\mathfrak{p}$-side results lean on structural hypotheses that the Ellentuck-type spaces do not satisfy.

Problem Statement

Determine what the topological Ramsey property forces upon the pair $(\mathfrak{p}_\mathcal{R},\mathfrak{t}_\mathcal{R})$. The source paper's Section 8 poses, verbatim, three questions:

(1) Is $\mathfrak{p}_\mathcal{R}\le\mathfrak{p}$ for every topological Ramsey space $\mathcal{R}$?

(2) Is there a topological Ramsey space with pseudointersection number different from its tower number?

(3) Is there a topological Ramsey space with pseudointersection number different from both $\omega_1$ and $\mathfrak{p}$?

Here 'topological Ramsey space' means a $\sigma$-closed topological Ramsey space $(\mathcal{R},\le,\le^*,r)$ as in Definition 23 of the source, and the invariants are those of Definition 24. Since always $\omega_1\le\mathfrak{p}_\mathcal{R}\le\mathfrak{t}_\mathcal{R}$: a negative answer to (1) is a space with $\mathfrak{p}_\mathcal{R}>\mathfrak{p}$; a positive answer to (2) is a space with $\mathfrak{p}_\mathcal{R}<\mathfrak{t}_\mathcal{R}$; and a positive answer to (3) is a space with $\mathfrak{p}_\mathcal{R}\notin\{\omega_1,\mathfrak{p}\}$, which, whenever $\mathfrak{p}_\mathcal{R}\le\mathfrak{p}$ and $\omega_1<\mathfrak{p}$, is exactly a value strictly between $\omega_1$ and $\mathfrak{p}$ (when $\mathfrak{p}=\omega_1$ it reduces to asking whether $\mathfrak{p}_\mathcal{R}$ can exceed $\mathfrak{p}$, i.e. to question (1)). The common objective of the three questions is a characterization of the possible values of $(\mathfrak{p}_\mathcal{R},\mathfrak{t}_\mathcal{R})$ over all $\sigma$-closed topological Ramsey spaces. No restriction to particular structural classes of spaces is part of the problem as stated.

References

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