Weakly tight MAD families in the $\mathfrak{b}<\mathfrak{s}$ regime and Sacks indestructibility

A maximal almost disjoint (MAD) family $\mathcal{A}$ of infinite subsets of $\omega$ is weakly tight if for every countable family of $\mathcal{I}(\mathcal{A})$-positive sets there is a member of $\mathcal{A}$ meeting infinitely many of them infinitely often. Raghavan and Steprāns (2012) constructed weakly tight families of size $\mathfrak{c}$ from the hypothesis $\mathfrak{s}_{\omega,\omega}\le\mathfrak{b}$, introducing the splitting-type invariant $\mathfrak{s}_{\omega,\omega}$ for this purpose, and posed a cluster of open questions. Mildenberger, Raghavan and Steprāns (2014) then proved $\mathfrak{s}_{\omega,\omega}=\mathfrak{s}$ in ZFC, which closes the invariant questions from the source (Questions 21 and 22 there) and makes $\mathfrak{s}\le\mathfrak{b}$ a ZFC sufficient condition for weakly tight families. This problem isolates the surviving open core of that question cluster. (1) Can $\mathfrak{a}$ replace $\mathfrak{b}$: must a weakly tight family exist whenever $\mathfrak{s}\le\mathfrak{a}<\aleph_\omega$? Since the $\mathfrak{s}\le\mathfrak{b}$ case is settled, the open regime is $\mathfrak{b}<\mathfrak{s}\le\mathfrak{a}$. (2) Can the regime $\mathfrak{b}<\mathfrak{s}$ consistently preclude weakly tight families altogether — conjecturally in a model of $\aleph_1=\mathfrak{b}<\mathfrak{s}=\aleph_2=\mathfrak{c}$ — and do weakly tight families exist in the known models of $\mathfrak{b}<\mathfrak{s}$? (3) Does the same construction technique strengthen to produce a Sacks-indestructible MAD family from $\mathfrak{s}\le\mathfrak{b}<\aleph_\omega$?

Background

Two infinite subsets of $\omega$ are almost disjoint (a.d.) if their intersection is finite; an a.d. family that is maximal under inclusion among a.d. families is a MAD family. For an a.d. family $\mathcal{A}$, $\mathcal{I}(\mathcal{A})$ denotes the ideal on $\omega$ generated by $\mathcal{A}$ together with the finite sets, and $\mathcal{I}^{+}(\mathcal{A})=\mathcal{P}(\omega)\setminus\mathcal{I}(\mathcal{A})$ is the family of positive sets. Hrušák and García Ferreira introduced weak tightness through the Katětov order on a.d. families ($\mathcal{A}\le_{K}\mathcal{B}$ iff some function $f:\omega\to\omega$ pulls every member of $\mathcal{I}(\mathcal{A})$ into $\mathcal{I}(\mathcal{B})$): $\mathcal{A}$ is weakly tight if for every countable $\{b_{n}:n\in\omega\}\subseteq\mathcal{I}^{+}(\mathcal{A})$ there is $a\in\mathcal{A}$ with $|b_{n}\cap a|=\omega$ for infinitely many $n$. This weakens tightness (also called $\aleph_{0}$-MAD or strongly MAD), which demands the same for all $n$ at once and is essentially equivalent to Cohen indestructibility of the MAD family. Weakly tight families are automatically MAD, are 'almost maximal' in the Katětov order, and are the existence counterpart of having few partitioners: a partitioner of $\mathcal{A}$ is a positive set $b$ such that every $a\in\mathcal{A}$ is almost included in $b$ or almost disjoint from $b$, and any a.d. family with an infinite pairwise disjoint family of partitioners fails weak tightness. Here $\mathfrak{s}$ is the splitting number (the least size of a family $\mathcal{S}\subseteq[\omega]^{\omega}$ such that every infinite $X\subseteq\omega$ is split by some $S\in\mathcal{S}$, meaning both $S\cap X$ and $X\setminus S$ are infinite), $\mathfrak{b}$ the bounding number (the least size of a family in $\omega^{\omega}$ unbounded under eventual domination), $\mathfrak{d}$ its dual dominating number, $\mathfrak{a}$ the almost disjointness number (the least size of a MAD family), and $\mathfrak{c}=2^{\aleph_{0}}$. A program going back to Erdős and Shelah's 1972 question whether a completely separable MAD family exists ($\mathcal{A}$ is completely separable if every $\mathcal{I}^{+}(\mathcal{A})$ set contains a member of $\mathcal{A}$) aims to construct MAD families with strong combinatorial properties in ZFC, whereas nearly all classical constructions need hypotheses of the form $\mathfrak{x}=\mathfrak{c}$. Shelah's breakthrough ('MAD saturated families and SANE player') gave completely separable MAD families by a three-case analysis comparing $\mathfrak{s}$ with $\mathfrak{a}$: $\mathfrak{s}<\mathfrak{a}$ in ZFC, and $\mathfrak{s}=\mathfrak{a}$ or $\mathfrak{a}<\mathfrak{s}$ under PCF-type hypotheses (statements about $\operatorname{cf}([\kappa]^{\omega},\subseteq)$ that hold below $\aleph_{\omega}$), yielding ZFC existence when $\mathfrak{c}<\aleph_{\omega}$. Raghavan and Steprāns adapted this machinery to weak tightness, comparing $\mathfrak{s}$ with $\mathfrak{b}$ instead. They introduced the invariant $\mathfrak{s}_{\omega,\omega}$: the least $\kappa$ such that there is $\{e_{\alpha}:\alpha<\kappa\}\subseteq[\omega]^{\omega}$ with the property that for every countable $\{b_{n}:n\in\omega\}\subseteq[\omega]^{\omega}$ some $\alpha$ satisfies $|b_{n}\cap e_{\alpha}|=\omega$ for infinitely many $n$ and $|b_{n}\setminus e_{\alpha}|=\omega$ for infinitely many $n$ — a countable-family strengthening of splitting. They proved in ZFC that a weakly tight family of size $\mathfrak{c}$ exists whenever $\mathfrak{s}_{\omega,\omega}\le\mathfrak{b}$, showed $\mathfrak{s}_{\omega,\omega}=\mathfrak{s}$ when $\mathfrak{s}<\mathfrak{b}$, and handled $\mathfrak{s}=\mathfrak{b}$ under a PCF-type principle $P(\kappa)$ true below $\aleph_{\omega}$, so weakly tight families exist whenever $\mathfrak{s}\le\mathfrak{b}<\aleph_{\omega}$. Mildenberger, Raghavan and Steprāns then proved $\mathfrak{s}_{\omega,\omega}=\mathfrak{s}$ outright in ZFC, so the hypothesis is now simply $\mathfrak{s}\le\mathfrak{b}$ with no PCF assumption; the same paper produced completely separable MAD families under $\mathfrak{s}\le\mathfrak{a}$. The resulting asymmetry is the subject of this problem: for complete separability both the $\mathfrak{b}$-side (e.g. $\mathfrak{b}=\mathfrak{d}$) and the $\mathfrak{a}$-side ($\mathfrak{s}\le\mathfrak{a}$) are settled, but for weak tightness only the $\mathfrak{b}$-side is, and nothing is known in the $\mathfrak{b}<\mathfrak{s}$ regime — whose consistency was first established by Shelah (his method is flexible enough to give $\mathfrak{a}=\mathfrak{b}<\mathfrak{s}$) and by Brendle via a c.c.c. iteration for $\mathfrak{b}<\mathfrak{a}=\mathfrak{s}$. A cautionary result of Guzmán shows that $\mathfrak{a}$-side hypotheses do not deliver weak tightness for free: under $\mathfrak{s}\le\mathfrak{a}$ there is a +-Ramsey MAD family (a Ramsey-type strengthening of the MAD property) that is not weakly tight. Sacks forcing (conditions are perfect subtrees of $2^{<\omega}$, ordered by inclusion) adds a new real of minimal degree; a MAD family is Sacks indestructible if it stays maximal after Sacks forcing, and this is the weakest indestructibility notion among forcings adding reals. Known constructions of Sacks indestructible MAD families of size $\mathfrak{c}$ use $\mathfrak{b}=\mathfrak{c}$ or $\operatorname{cov}(\mathcal{M})=\mathfrak{c}$ (the least number of meager sets covering $\mathbb{R}$; if $\mathfrak{a}<\mathfrak{c}$ then every MAD family of size $\mathfrak{a}$ is automatically Sacks indestructible), and whether one exists in ZFC is a well-known open problem of Hrušák. The Raghavan-Steprāns machinery of node ideals $\mathcal{I}_{\eta}$ on a binary tree $2^{<\kappa}$ — controlling which countable sequences of infinite sets may be associated with each node, so that sets associated with incomparable nodes are automatically a.d. — is the natural candidate for such constructions from hypotheses of the form $\mathfrak{s}\le\mathfrak{b}<\aleph_{\omega}$.

Problem Statement

The problem is the surviving open core of the question cluster posed in Section 4 of Raghavan and Steprāns, 'On weakly tight families' (2012), after the invariant questions from that paper were closed by Mildenberger, Raghavan and Steprāns (2014). Facets A and B concern completing the case analysis for the existence of weakly tight families — deciding whether their existence is governed by $\mathfrak{b}$, so that $\mathfrak{b}<\mathfrak{s}$ is a genuine barrier, or by $\mathfrak{a}$, which would match the situation for completely separable MAD families — while Facet C asks whether the Raghavan-Steprāns construction technique strengthens from weak tightness to Sacks indestructibility. The source poses these as separate open problems (Question 23 and Conjectures 24-25 of Section 4). The three facets are:

Facet A (Question 23 of the source). Suppose $\mathfrak{s}\le\mathfrak{a}<\aleph_{\omega}$. Since weakly tight families of size $\mathfrak{c}$ are now known to exist in ZFC whenever $\mathfrak{s}\le\mathfrak{b}$, the open case is $\mathfrak{b}<\mathfrak{s}\le\mathfrak{a}$. Must a weakly tight family exist under this hypothesis? Equivalently: can $\mathfrak{a}$ replace $\mathfrak{b}$ in the hypothesis of the Raghavan-Steprāns existence theorem?

Facet B (Conjecture 24 of the source, with the adjacent model questions). Is there a model of $\aleph_{1}=\mathfrak{b}<\mathfrak{s}=\aleph_{2}=\mathfrak{c}$ in which no weakly tight families exist? More broadly, do weakly tight families exist in the known consistency models of $\mathfrak{b}<\mathfrak{s}$ — Shelah's original construction (which can be arranged with $\mathfrak{a}=\mathfrak{b}<\mathfrak{s}$) and Brendle's c.c.c. iteration for $\mathfrak{b}<\mathfrak{a}=\mathfrak{s}$ — and, if so, of which sizes? (The source notes that from CH a weakly tight family can be made to survive the relevant iteration, but will not have size $\mathfrak{c}$ there, and it is unknown whether any of size $\mathfrak{s}$ exist in that model.)

Facet C (Conjecture 25 of the source). If $\mathfrak{s}\le\mathfrak{b}<\aleph_{\omega}$, is there a Sacks-indestructible MAD family? (The source notes that for proving this conjecture one may assume $\mathfrak{a}=\mathfrak{c}$.)

No restriction on family size, model class, or method is intended beyond what is explicitly stated; in particular, 'weakly tight family' always means the source's definition for a.d. families of infinite subsets of $\omega$, and any claimed nonexistence result must rule out weakly tight families of every size, not only of size $\mathfrak{c}$.

References

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