Is the dual reaping number $\mathfrak{r}_d$ at most the pair-splitting number $\mathfrak{s}_{\mathrm{pair}}$?

The dual reaping number $\mathfrak{r}_d$ is the least size of a family of infinite partitions of $\omega$ (ordered by "almost coarser") that no single partition dual-splits; the pair-splitting number $\mathfrak{s}_{\mathrm{pair}}$ is the least size of a family of infinite subsets of $\omega$ that pair-splits every unbounded family of $2$-element subsets of $\omega$. The pair-splitting number was introduced in 2010 by Hiroaki Minami precisely in order to calibrate the dual partition invariants $\mathfrak{s}_d,\mathfrak{r}_d$ (the dual reaping number itself goes back in print to Cichoń, Majcher and Węglorz, 1992); $\mathfrak{s}_{\mathrm{pair}}$ lies between the classical splitting number $\mathfrak{s}$ and the uniformity numbers $\operatorname{non}(\mathcal{M}),\operatorname{non}(\mathcal{N})$ of the meager and null ideals, which also bound $\mathfrak{r}_d$ from above. The basic ZFC comparison between them was explicitly left open (Question 3 of Minami's paper; verbatim as "It is not known whether $\mathfrak{r}_d\le\mathfrak{s}_{\mathrm{pair}}$ or not" and as Question 4.8(1) in the companion paper of Hrušák–Meza-Alcántara–Minami): must every dual-reaping family of partitions be matched in size by a pair-splitting family, i.e. does $\mathfrak{r}_d\le\mathfrak{s}_{\mathrm{pair}}$ hold in ZFC? The follow-up literature — including the 2024 generalization of pair-splitting by Shelah–Steprāns and Weinert's 2026 comparative study of cardinal-characteristic inequalities, whose Hasse diagram records no $\mathfrak{r}_d$–$\mathfrak{s}_{\mathrm{pair}}$ edge — contains neither a ZFC proof nor a consistency counterexample, so the question remains open.

Background

A cardinal characteristic of the continuum is an infinite cardinal between $\aleph_1$ and $\mathfrak{c}=2^{\aleph_0}$ measuring the least size of a family of simple objects on $\omega$ needed to carry out some infinitary combinatorial task. Two of the oldest are the splitting number $\mathfrak{s}$ — the least size of a family $\mathcal{S}\subseteq[\omega]^\omega$ such that every infinite $Y\subseteq\omega$ is split by some $X\in\mathcal{S}$ (both $X\cap Y$ and $Y\setminus X$ infinite) — and the reaping (unsplitting) number $\mathfrak{r}$, the least size of a family $\mathcal{R}\subseteq[\omega]^\omega$ no single $X$ splits. ZFC (the standard axioms of set theory) does not fix their values, so they are studied through ZFC inequalities and forcing consistency results, organized in van Douwen's diagram and Cichoń's diagram; the latter is built from the additivity, covering, uniformity and cofinality cardinals of the meager ideal $\mathcal{M}$ and the null ideal $\mathcal{N}$, alongside the bounding number $\mathfrak{b}$ and the dominating number $\mathfrak{d}$ for eventual domination on $\omega^\omega$.

A line of work going back to Carlson–Simpson's dual Ramsey theorem replaces subsets of $\omega$ by partitions. Let $(\omega)^\omega$ be the set of partitions of $\omega$ into infinitely many nonempty pieces. A partition $R$ almost refines a partition $X$ if all but finitely many blocks of $R$ are, modulo finite error, contained in blocks of $X$; write $X\leq^*Y$ ("$X$ is almost coarser than $Y$") when $Y$ almost refines $X$. The meet $X\wedge Y=\{x\cap y:x\in X,y\in Y\}\setminus\{\emptyset\}$ is the common refinement, and $X,Y$ are compatible if $X\wedge Y$ is an infinite partition. A partition $S$ dual-splits $X$ if $S$ and $X$ are compatible (equivalently, some infinite partition almost refines both) but $S$ does not almost refine $X$. The dual splitting number $\mathfrak{s}_d$ is the least size of a family $\mathcal{S}\subseteq(\omega)^\omega$ such that every partition is dual-split by some member of $\mathcal{S}$; the dual reaping number $\mathfrak{r}_d$, introduced in print by Cichoń, Majcher and Węglorz (1992), is the least size of a family $\mathcal{R}\subseteq(\omega)^\omega$ such that no single partition dual-splits every member of $\mathcal{R}$. This "dual van Douwen diagram" was developed by Cichoń–Krawczyk–Majcher-Iwanow–Węglorz and by Halbeisen. Kamo (proofs published in Minami's 2010 paper) proved $\mathfrak{r}_d\le\mathfrak{d}$ and $\mathfrak{s}_d\ge\mathfrak{b}$, and Brendle (Proposition 6 of the same paper) proved $\mathfrak{r}_d\le\operatorname{non}(\mathcal{M}),\operatorname{non}(\mathcal{N})$; Minami proved the consistency of $\mathfrak{r}_d>\mathfrak{b}$, $\mathfrak{s}_d<\mathfrak{d}$, $\mathfrak{s}_d<\mathfrak{r}$, $\mathfrak{r}_d<\operatorname{add}(\mathcal{M})$ and $\mathfrak{s}_d>\operatorname{cof}(\mathcal{M})$, the last two via Hechler iterations.

To calibrate $\mathfrak{s}_d$ and $\mathfrak{r}_d$ further, Minami introduced pair-splitting and pair-reaping. A family $\mathcal{A}\subseteq[\omega]^2$ is unbounded if for every $k<\omega$ some $a\in\mathcal{A}$ is disjoint from $k$. An infinite set $X\subseteq\omega$ pair-splits $\mathcal{A}$ if infinitely many $a\in\mathcal{A}$ meet both $X$ and its complement; $\mathfrak{s}_{\mathrm{pair}}$ is the least size of a family $\mathcal{S}\subseteq[\omega]^\omega$ pair-splitting every unbounded $\mathcal{A}\subseteq[\omega]^2$, and $\mathfrak{r}_{\mathrm{pair}}$ is the least size of a family of unbounded subsets of $[\omega]^2$ such that every infinite $X$ fails to pair-split some member. Minami proved $\mathfrak{s}\le\mathfrak{s}_{\mathrm{pair}}\le\operatorname{non}(\mathcal{M}),\operatorname{non}(\mathcal{N})$ and $\operatorname{cov}(\mathcal{M}),\operatorname{cov}(\mathcal{N})\le\mathfrak{r}_{\mathrm{pair}}\le\mathfrak{r},\mathfrak{s}_d$, and the consistency of $\mathfrak{s}_{\mathrm{pair}}<\operatorname{add}(\mathcal{M})$ and $\mathfrak{r}_{\mathrm{pair}}>\operatorname{cof}(\mathcal{M})$. The companion paper of Hrušák–Meza-Alcántara–Minami identified $\mathfrak{s}_{\mathrm{pair}}$ with the covering number $\operatorname{cov}^*$ of $\mathcal{G}_{FC}$, the $F_\sigma$ ideal of finitely chromatic graphs on $\omega$ (edge sets with finite chromatic number), proved $\mathfrak{s}_{\mathrm{pair}}=\mathfrak{s}_n$ and $\mathfrak{r}_{\mathrm{pair}}=\mathfrak{r}_n$ (the $n$-element analogues for $n\ge2$ and $n\ge3$ respectively), and recorded the consistency of $\mathfrak{r}_{\mathrm{pair}}<\mathfrak{b}$ and $\mathfrak{s}_{\mathrm{pair}}>\mathfrak{d}$. Later work sharpened the pair side: Shelah–Steprāns (2024) generalized pair-splitting to splitting finite sets of prescribed sizes into $k$ pieces (cardinals $\mathfrak{s}_{k,f}$), and Weinert (2026) proved $\mathfrak{e}_{ubd}\le\mathfrak{s}_{\mathrm{pair}}\le\mathfrak{ß}$, where $\mathfrak{e}_{ubd}$ is the unbounded evasion number and $\mathfrak{ß}$ the subseries number.

One basic comparison is conspicuously missing. The mirror-image cross-comparison $\mathfrak{r}_{\mathrm{pair}}\le\mathfrak{s}_d$ is a theorem, and both $\mathfrak{r}_d$ and $\mathfrak{s}_{\mathrm{pair}}$ are bounded above by $\operatorname{non}(\mathcal{M})$ and $\operatorname{non}(\mathcal{N})$; but no ZFC inequality between $\mathfrak{r}_d$ and $\mathfrak{s}_{\mathrm{pair}}$ is known, and no model with $\mathfrak{s}_{\mathrm{pair}}<\mathfrak{r}_d$ has been constructed. Both 2010 papers mark exactly this edge with a question mark, and it is still absent from the most recent comparative diagram (Weinert, 2026).

Problem Statement

Determine in ZFC the relation between the dual reaping number $\mathfrak{r}_d$ and the pair-splitting number $\mathfrak{s}_{\mathrm{pair}}$. Specifically: is $\mathfrak{r}_d\le\mathfrak{s}_{\mathrm{pair}}$ a theorem of ZFC — equivalently, can every pair-splitting family $\mathcal{S}\subseteq[\omega]^\omega$ be converted into a family of partitions of $\omega$ of size at most $|\mathcal{S}|$ that no single partition dual-splits? The question is whether $\mathfrak{r}_d\le\mathfrak{s}_{\mathrm{pair}}$ is provable in ZFC, or whether instead there is a model of ZFC in which the strict inequality $\mathfrak{s}_{\mathrm{pair}}<\mathfrak{r}_d$ holds, refuting the inequality. For calibration: the mirror-image cross-comparison $\mathfrak{r}_{\mathrm{pair}}\le\mathfrak{s}_d$ is already known, so the question is the last missing basic comparison between the dual partition cardinals and the pair cardinals.

References

1. Hiroaki Minami. Around splitting and reaping for partitions of ω. Archive for Mathematical Logic 49(4):501–518, 2010. DOI: 10.1007/s00153-010-0184-9. https://doi.org/10.1007/s00153-010-0184-9
2. Michael Hrušák, David Meza-Alcántara, Hiroaki Minami. Pair-splitting, pair-reaping and cardinal invariants of Fσ-ideals. Journal of Symbolic Logic 75(2):661–677, 2010. DOI: 10.2178/jsl/1268917498. https://doi.org/10.2178/jsl/1268917498
3. Thilo Weinert. Calligraphy Concerning Casually Compiled Cardinal Characteristic Comparisons. Preprint, arXiv:2407.09630 (v2, 18 March 2026). arXiv ID: 2407.09630. https://arxiv.org/abs/2407.09630
4. Saharon Shelah, Juris Steprāns. Some variations on the splitting number. Annals of Pure and Applied Logic 175(1, part B):103321, 2024. DOI: 10.1016/j.apal.2023.103321. https://doi.org/10.1016/j.apal.2023.103321
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