Existence of a completely separable MAD family in ZFC (the Erdős–Shelah problem)

A MAD family (maximal almost disjoint family) of infinite subsets of $\omega$ is completely separable if every set that is not almost covered by finitely many members of the family contains one of its members. Whether a completely separable MAD family can be constructed in ZFC, without any additional hypothesis, is a problem open since Erdős and Shelah posed it in 1972. Existence is provable in ZFC under $\mathfrak{s}\le\mathfrak{a}$ (Mildenberger–Raghavan–Steprāns, 2014, building on Shelah's 2011 result making existence 'almost a theorem of ZFC'), and in the complementary regime $\mathfrak{a}<\mathfrak{s}$ only under the PCF-type hypothesis $P(\mathfrak{s},\mathfrak{a})$, which holds automatically whenever $\mathfrak{s}<\aleph_\omega$; Shelah proved that non-existence would force a PCF-related situation of large consistency strength. The problem asks for a ZFC resolution: a construction from the axioms alone — the only uncovered case being $\mathfrak{a}<\mathfrak{s}$ with failure of $P(\mathfrak{s},\mathfrak{a})$ — or a model of ZFC containing no completely separable MAD family.

Background

Two infinite subsets of $\omega$ are almost disjoint if their intersection is finite, and a family $\mathcal{A}\subseteq[\omega]^\omega$ is almost disjoint (a.d.) if its members are pairwise almost disjoint; a MAD family (maximal almost disjoint family) is an infinite a.d. family that cannot be enlarged. Write $X\subseteq^*Y$ if $X\setminus Y$ is finite. For an a.d. family $\mathcal{A}$, let $I(\mathcal{A})$ be the ideal on $\omega$ generated by $\mathcal{A}$: $X\in I(\mathcal{A})$ iff $X\subseteq^*A_0\cup\dots\cup A_n$ for some $A_0,\dots,A_n\in\mathcal{A}$; and let $I^+(\mathcal{A})$ denote $\mathcal{P}(\omega)\setminus I(\mathcal{A})$.

Following Hechler (1971) and Erdős and Shelah (1972), an a.d. family $\mathcal{A}$ is completely separable if every $X\in I^+(\mathcal{A})$ contains some $A\in\mathcal{A}$; equivalently, every $X\subseteq\omega$ is either almost covered by finitely many members of $\mathcal{A}$ or contains a member of $\mathcal{A}$. An infinite completely separable a.d. family is automatically MAD. Such families are a basic interface between combinatorics and topology: the Isbell–Mrówka space $\Psi(\mathcal{A})$ is built from $\mathcal{A}$ by keeping the points of $\omega$ isolated and adding one point for each member of $\mathcal{A}$ whose neighborhoods trace that member, so combinatorial properties of $\mathcal{A}$ become topological properties (Fréchet behavior, sequential compactness, pseudocompactness) of $\Psi(\mathcal{A})$; and Lávička and Verner showed that the existence of a completely separable MAD family implies completeness of the modal logic S4.1.2 for the Čech–Stone compactification $\beta\omega$ of the integers.

The problem sits among the cardinal invariants of the continuum, the cardinal numbers between $\aleph_1$ and $\mathfrak{c}=2^{\aleph_0}$ that measure combinatorial properties of $\mathcal{P}(\omega)$. Relevant here are: the splitting number $\mathfrak{s}$, the least size of a family $S\subseteq\mathcal{P}(\omega)$ such that every infinite $X\subseteq\omega$ is split by some member of $S$ (both $X\cap S$ and $X\setminus S$ infinite); the almost-disjointness number $\mathfrak{a}$, the least size of a MAD family; the bounding number $\mathfrak{b}$ and dominating number $\mathfrak{d}$ for eventual domination in $\omega^\omega$; and the variant $\mathfrak{s}_{\omega,\omega}$, the least size of a family $F\subseteq\mathcal{P}(\omega)$ such that for every countable $\{a_n:n\in\omega\}\subseteq[\omega]^\omega$ some $x\in F$ has $x\cap a_n$ infinite for infinitely many $n$ and $(\omega\setminus x)\cap a_n$ infinite for infinitely many $n$. ZFC does not determine the relative order of $\mathfrak{a}$ and $\mathfrak{s}$.

Completely separable a.d. families that are not maximal — in the weaker sense of complete separability used by Galvin and Simon, where the test sets are $I^{++}(\mathcal{A})=\{X\subseteq\omega:|\{A\in\mathcal{A}:|X\cap A|=\omega\}|\ge\omega\}$ rather than the $I^+(\mathcal{A})$ of the definition above, the two conventions agreeing on MAD families — are easy to build in ZFC: Galvin and Simon constructed a completely separable nowhere MAD family (nowhere MAD: every $I^+(\mathcal{A})$ set contains an infinite set almost disjoint from all of $\mathcal{A}$) — so maximality is the entire difficulty. Existence of completely separable MAD families has long been known from additional hypotheses: from $\mathfrak{a}=\mathfrak{c}$, and from $\mathfrak{b}=\mathfrak{d}$, $\mathfrak{s}=\aleph_1$, or $\mathfrak{d}\le\mathfrak{a}$ (results of Balcar, Dočkálková and Simon, recounted in the survey of Hrušák and Simon and in Mildenberger–Raghavan–Steprāns). Shelah (2011) then made existence 'almost a theorem of ZFC': his construction works in ZFC alone when $\mathfrak{s}<\mathfrak{a}$; when $\mathfrak{s}=\mathfrak{a}$ it needs the principle $U(\mathfrak{s})$; and when $\mathfrak{a}<\mathfrak{s}$ it needs the principle $P(\mathfrak{s},\mathfrak{a})$. Here $U(\kappa)$ and $P(\kappa,\lambda)$ are PCF-type guessing principles (PCF, the theory of possible cofinalities, is Shelah's theory of singular cardinals): $U(\kappa)$ asks for a sequence $\langle u_\alpha:\omega\le\alpha<\kappa\rangle$ of countable sets $u_\alpha\subseteq\alpha$ such that every $X\in[\kappa]^\kappa$ meets some $u_\alpha$ infinitely often, and $P(\kappa,\lambda)$, for $\omega<\lambda<\kappa$, asks for such a sequence such that every bounded $X\subseteq\kappa$ of order type $\lambda$ is met infinitely often by some $u_\alpha$ with $\alpha<\sup X$. Both principles hold whenever $\mathfrak{s}<\aleph_\omega$, so existence is a theorem of ZFC whenever $\mathfrak{c}<\aleph_\omega$. Shelah also proved that a model of ZFC containing no completely separable MAD family must satisfy a PCF-related situation of large consistency strength, and asked whether all uses of PCF-type hypotheses can be eliminated.

Mildenberger, Raghavan and Steprāns (2014) proved in ZFC that $\mathfrak{s}=\mathfrak{s}_{\omega,\omega}$ and used it to unify and strengthen Shelah's first two regimes: a completely separable MAD family exists in ZFC whenever $\mathfrak{s}\le\mathfrak{a}$, with no PCF hypothesis. (Their argument uses that the splitting number never has countable cofinality — a ZFC theorem, since in fact $\mathrm{cf}(\mathfrak{s})\ge\mathfrak{t}$, the tower number, the least length of a $\supseteq^*$-decreasing tower of subsets of $\omega$ with no infinite pseudointersection — although $\mathfrak{s}$ can consistently be singular of uncountable cofinality, as Dow and Shelah showed.) The single regime covered by no known result is therefore $\mathfrak{a}<\mathfrak{s}$ with failure of $P(\mathfrak{s},\mathfrak{a})$, which can occur only when $\mathfrak{s}\ge\aleph_\omega$. Recent specialist literature (Lávička–Verner 2022; Corral–Rodrigues 2023; Corral–Guzmán–López-Callejas 2024) uniformly records the ZFC existence question as open, and the analogous question for MAD families of block subspaces of a countable-dimensional vector space over a countable field was answered affirmatively in ZFC by Yung, so the classical set-theoretic case is the genuinely open one.

Problem Statement

The problem is the question of Erdős and Shelah (1972), in the precise form in which Mildenberger–Raghavan–Steprāns (2014) left it open and which the subsequent literature still records as open: does ZFC prove that there exists a completely separable MAD family $\mathcal{A}\subseteq[\omega]^\omega$, i.e. an infinite a.d. family such that every $X\in I^+(\mathcal{A})$ contains some $A\in\mathcal{A}$?

An affirmative answer is a construction (or other proof) of such a family from the ZFC axioms alone, with no additional set-theoretic hypothesis. By the known partial results, this reduces to the single uncovered regime: $\mathfrak{a}<\mathfrak{s}$ together with failure of the PCF-type principle $P(\mathfrak{s},\mathfrak{a})$, which can occur only when $\mathfrak{s}\ge\aleph_\omega$. A negative answer establishes, from some explicitly stated consistent hypothesis, the relative consistency of ZFC with 'there is no completely separable MAD family'; by Shelah's theorem such a model necessarily realizes a PCF-related situation of substantial consistency strength, so the hypothesis is expected to carry that strength. Answers that assume $\mathfrak{s}\le\mathfrak{a}$, or $\mathfrak{a}<\mathfrak{s}$ with $P(\mathfrak{s},\mathfrak{a})$, or $\mathfrak{c}<\aleph_\omega$, reprove known results and do not resolve the problem.

References

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10. C. Corral and V. de O. Rodrigues, Fin-intersecting MAD families, preprint, arXiv:2212.01484 (2022). https://arxiv.org/abs/2212.01484
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12. J. Brendle, O. Guzmán, M. Hrušák, and D. Raghavan, Combinatorial properties of MAD families, Canadian Journal of Mathematics (2025), pp. 1–69 (online first). DOI: 10.4153/S0008414X25101879, arXiv:2206.14936, https://doi.org/10.4153/S0008414X25101879
13. C. Yung, Full mad families of vector spaces and two local Ramsey theories, preprint, arXiv:2503.24207 (2025). https://arxiv.org/abs/2503.24207