Splitting versus closed almost disjointness: does $\mathfrak{s} = \aleph_1$, or at least $\mathfrak{s}_\omega = \aleph_1$, imply $\mathfrak{a}_{closed} = \aleph_1$?

An almost disjoint (a.d.) family $\mathcal{A} \subseteq [\omega]^\omega$ is maximal (MAD) if no infinite set is a.d. from every member. Brendle and Khomskii introduced the closed almost disjointness number $\mathfrak{a}_{closed}$ — the least number of closed a.d. subsets of $[\omega]^\omega$ (with the subspace topology inherited from the Cantor space) whose union is a MAD family — and Brendle and Raghavan ('Bounding, splitting, and almost disjointness', 2014) proved that strong splitting hypotheses force it to be $\aleph_1$: a club-splitting sequence of partitions of length $\omega_1$ (and a fortiori $\mathfrak{d} = \aleph_1$) yields $\mathfrak{a}_{closed} = \aleph_1$, while conversely Con($\aleph_1 = \mathfrak{b} < \mathfrak{a}_{closed} = \aleph_2$) holds. This problem records the three questions they left open (their Questions 46–48): (i) does $\mathfrak{s} = \aleph_1$, or at least the stronger hypothesis $\mathfrak{s}_\omega = \aleph_1$, already imply $\mathfrak{a}_{closed} = \aleph_1$? (ii) is it consistent that $\mathfrak{s} = \aleph_1$ (or $\mathfrak{s}_\omega = \aleph_1$) yet no club-splitting sequence of length $\omega_1$ exists? (iii) does a tail-splitting (club-splitting) sequence of length $\kappa$ imply a tail-splitting (club-splitting) sequence of partitions of length $\kappa$? These three questions remain open. Raghavan and Steprāns (2023) pose further open questions about $\mathfrak{a}_{closed}$ and observe that the techniques currently available for making almost-disjointness-type invariants large also make $\mathfrak{s}$ large.

Background

An infinite family $\mathcal{A} \subseteq [\omega]^\omega$ is almost disjoint (a.d.) if $a \cap b$ is finite for distinct $a, b \in \mathcal{A}$, and maximal almost disjoint (MAD) if no infinite subset of $\omega$ is almost disjoint from every member of $\mathcal{A}$. The almost disjointness number $\mathfrak{a}$ — the least size of a MAD family — is one of the classical cardinal characteristics of the continuum: uncountable cardinals at most $2^{\aleph_0}$ measuring combinatorial properties of subsets of $\omega$, whose provable inequalities and consistent separations form a central thread of combinatorial set theory. Two neighbors used below: the bounding number $\mathfrak{b}$ is the least size of a family of functions $\omega \to \omega$ unbounded under eventual domination ($f \leq^* g$ iff $f(n) \leq g(n)$ for all but finitely many $n$), and the dominating number $\mathfrak{d}$ is the least size of a family eventually dominating every such function. Identifying $[\omega]^\omega$ with a subspace of the Cantor space $2^\omega$ (the set of infinite binary sequences, i.e. $\mathcal{P}(\omega)$, with the product topology), one can ask how definable the members of a MAD family must be. Brendle and Khomskii isolated two refinements of $\mathfrak{a}$: $\mathfrak{a}_{closed}$, the least number of closed a.d. subsets of $[\omega]^\omega$ (with the subspace topology) whose union is a MAD family, and the Borel almost-disjointness number $\mathfrak{a}_B$, the same with 'closed' replaced by 'Borel'. Trivially $\aleph_1 \leq \mathfrak{a}_B \leq \mathfrak{a}_{closed} \leq \mathfrak{a}$. Brendle–Khomskii established $\mathfrak{t} \leq \mathfrak{a}_B$ (a theorem of Raghavan, published in their paper), where $\mathfrak{t}$ is the tower number (the least length of a $\subseteq^*$-decreasing tower of infinite sets having no infinite pseudointersection; $a \subseteq^* b$ means $a \setminus b$ is finite, and a pseudointersection is an infinite set $\subseteq^*$-included in every tower member), and that $\mathfrak{a}_B = \aleph_1 < \mathfrak{b}$ holds in the iterated Hechler model (adding $\aleph_2$ Hechler reals by a finite-support iteration — each step adds a dominating real, and ccc (countable chain condition) means every antichain of the forcing is countable — over a model of CH, the continuum hypothesis $2^{\aleph_0} = \aleph_1$), and Raghavan–Shelah showed that $\mathfrak{d} = \aleph_1$ implies $\mathfrak{a}_{closed} = \aleph_1$. Brendle–Raghavan proved the complementary consistency Con($\aleph_1 = \mathfrak{b} < \mathfrak{a}_{closed} = \aleph_2$), so $\mathfrak{b}$ and $\mathfrak{a}_{closed}$ are independent. The phenomenon making $\mathfrak{a}_{closed}$ delicate is code reinterpretation: when a forcing extension adds reals, a ground-model code for a closed a.d. set comes to denote a larger set, so destroying a candidate witness $\bigcup_{\alpha < \kappa} X_\alpha$ requires adding a real almost disjoint from every member of every $X_\alpha$ even after this enlargement — harder than the corresponding task for $\mathfrak{a}$. A set $x \subseteq \omega$ splits $a \in [\omega]^\omega$ if both $a \cap x$ and $a \setminus x$ are infinite; the splitting number $\mathfrak{s}$ is the least size of a family splitting every infinite $a$. Brendle–Raghavan introduced the $\omega$-splitting number $\mathfrak{s}_\omega$: the least size of $\mathcal{F} \subseteq \mathcal{P}(\omega)$ such that every countable $\{a_n : n \in \omega\} \subseteq [\omega]^\omega$ has a single $x \in \mathcal{F}$ splitting all $a_n$ simultaneously; clearly $\mathfrak{s} \leq \mathfrak{s}_\omega$. They organized the strongest forms as sequences: for a regular cardinal $\kappa$, $\bar{A} = \langle a_\alpha : \alpha < \kappa \rangle$ is club-splitting if for every $b \in [\omega]^\omega$ the set $\{\alpha : a_\alpha \text{ splits } b\}$ contains a club (closed and unbounded) subset of $\kappa$, and tail-splitting if every $b$ is split by all sufficiently large $\alpha$; tail-splitting implies club-splitting, and a club-splitting sequence of length $\kappa$ implies $\mathfrak{s}_\omega \leq \kappa$. The partition versions are stronger hypotheses (each yields a plain splitting sequence, and whether the converse holds is item (iii) of this problem): a matrix $\langle a_{\alpha,n} : \alpha < \kappa, n < \omega\rangle$ whose every row $\{a_{\alpha,n} : n < \omega\}$ is pairwise disjoint, with all members of row $\alpha$ splitting $b$ for club-many (respectively, all sufficiently large) $\alpha$. Brendle–Raghavan proved: a club-splitting sequence of partitions of length $\omega_1$ yields, via an intermediate technical notion of a splitting sequence of partitions over countable models, a closed MAD family, hence $\mathfrak{a}_{closed} = \aleph_1$; $\mathfrak{d} = \aleph_1$ yields a tail-splitting sequence of partitions of length $\omega_1$; and it is consistent that a club-splitting family of size $\aleph_1$ exists while no tail-splitting family of size $\aleph_1$ does (so tail-splitting is strictly stronger than club-splitting). What remains unknown is how the weakest hypotheses in this chain relate to the conclusion — the subject of this problem. (These sequence notions also govern polarized partition relations such as $\binom{\omega_1}{\omega} \to \binom{\omega_1}{\omega}^{1,1}_2$ — the statement that for every 2-coloring of the pairs in $\omega_1 \times \omega$ there are uncountably many $\alpha$ and infinitely many $n$ on which the coloring is constant, a two-dimensional Ramsey-type statement studied by Garti–Shelah.)

Problem Statement

The problem is to complete the ZFC implication diagram relating the splitting-side hypotheses to the closed almost disjointness number, by settling the three gaps that Brendle–Raghavan (Questions 46–48 of 'Bounding, splitting, and almost disjointness') explicitly left open. Throughout, all notions are as defined in the background, and the ambient theory is ZFC.

(i) Does $\mathfrak{s} = \aleph_1$ — or at least the stronger hypothesis $\mathfrak{s}_\omega = \aleph_1$ — imply $\mathfrak{a}_{closed} = \aleph_1$? Concretely, a resolution of (i) either proves in ZFC that a splitting family of size $\aleph_1$ — or, for the variant with the stronger hypothesis, an $\omega$-splitting family of size $\aleph_1$ — yields a MAD family that is a union of $\aleph_1$ closed a.d. sets, or produces a model of ZFC in which $\mathfrak{s} = \aleph_1$ (or even $\mathfrak{s}_\omega = \aleph_1$) while $\mathfrak{a}_{closed} > \aleph_1$.

(ii) Is it consistent that $\mathfrak{s} = \aleph_1$, or even $\mathfrak{s}_\omega = \aleph_1$, yet there is no club-splitting sequence of length $\omega_1$? (A club-splitting sequence of length $\omega_1$ implies $\mathfrak{s}_\omega = \aleph_1$; the question is whether the converse can consistently fail, i.e. whether club-splitting is strictly stronger than $\mathfrak{s}_\omega = \aleph_1$.)

(iii) Does the existence of a tail-splitting sequence of length $\kappa$ imply the existence of a tail-splitting sequence of partitions of length $\kappa$? And the analogous question for club-splitting. (The partition versions are what Brendle–Raghavan's route from splitting to $\mathfrak{a}_{closed} = \aleph_1$ actually consumes; it is unknown whether they are equivalent to the plain versions.)

Each item that bundles two hypotheses or two splitting notions — (i) and (ii) with $\mathfrak{s} = \aleph_1$ versus the stronger $\mathfrak{s}_\omega = \aleph_1$, and (iii) with tail-splitting versus club-splitting — is a pair of independently resolvable sub-claims: determining one variant need not determine the other (they are tied only when a single witness settles both, as a model with $\mathfrak{s}_\omega = \aleph_1$ and $\mathfrak{a}_{closed} > \aleph_1$ does for the two variants of (i)). The three questions are the remaining gaps in one chain — tail-splitting partitions $\Rightarrow$ tail-splitting $\Rightarrow$ club-splitting $\Rightarrow \mathfrak{s}_\omega = \aleph_1 \Rightarrow \mathfrak{s} = \aleph_1$, with club-splitting partitions of length $\omega_1 \Rightarrow \mathfrak{a}_{closed} = \aleph_1$ — so together they ask exactly: how weak can a splitting hypothesis be and still force a small closed almost disjointness number?

References

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