The dual-reaping number versus the pair-splitting number, and star-uniformity of Borel ideals versus the cofinality of the null ideal (Question 4.8 of Hrušák–Meza-Alcántara–Minami, 2010)

Two questions asked as Question 4.8 of Hrušák–Meza-Alcántara–Minami (J. Symbolic Logic 75 (2010), 661–677). Part (1): in ZFC, is the dual reaping number $\mathfrak{r}_d$ — the reaping number of the lattice of infinite partitions of $\omega$ ordered by “almost coarser” — at most the pair-splitting number $\mathfrak{s}_{pair}$ , the least size of a family $\mathcal{S} \subseteq [\omega]^\omega$ such that every unbounded family $\mathcal{A} \subseteq [\omega]^2$ has infinitely many of its 2-element sets crossing some $X \in \mathcal{S}$ ? Part (2): in ZFC, is the star-uniformity $\mathrm{non}^*(\mathcal{I})$ of every Borel ideal $\mathcal{I}$ on $\omega$ bounded by the cofinality $\mathrm{cof}(\mathcal{N})$ of the Lebesgue-null ideal? Both parts remain open: no ZFC comparison between $\mathfrak{r}_d$ and $\mathfrak{s}_{pair}$ is known, and neither a ZFC proof of $\mathrm{non}^*(\mathcal{I}) \le \mathrm{cof}(\mathcal{N})$ for all Borel ideals nor a Borel ideal violating it in some model of ZFC is known. The source paper settles the $F_\sigma$ case of part (2) (Corollary 4.6 gives $\mathrm{non}^*(\mathcal{I}) \le \mathfrak{l}$ for $F_\sigma$ ideals, and the paper’s summary diagram records $\mathfrak{l} \le \mathrm{cof}(\mathcal{N})$ ) and proves the surrounding bounds, but explicitly states that $\mathfrak{r}_d \le \mathfrak{s}_{pair}$ is unknown.

Background

Cardinal invariants of the continuum are cardinal numbers between $\aleph_1$ and $\mathfrak{c} = 2^{\aleph_0}$ that measure how large a family of subsets of $\omega$ (or of reals) must be to achieve some combinatorial goal. Two classical examples: 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 $Y \cap X$ and $Y \setminus X$ infinite); and the reaping number $\mathfrak{r}$ , the least size of an unsplittable family, i.e. $\mathcal{R} \subseteq [\omega]^\omega$ such that every $X \subseteq \omega$ fails to split some $R \in \mathcal{R}$ (equivalently, some $R \in \mathcal{R}$ satisfies $R \subseteq^* X$ or $R \cap X =^* \emptyset$ , where $\subseteq^*$ and $=^*$ mean agreement up to finitely many exceptions). Together with the bounding number $\mathfrak{b}$ (the least size of a family of functions in $\omega^\omega$ no one of which eventually dominates all the others) and the dominating number $\mathfrak{d}$ (the least size of a family such that every function in $\omega^\omega$ is eventually dominated by one of its members), and the additivity/covering/uniformity/cofinality cardinals of the meager ideal $\mathcal{M}$ and the null ideal $\mathcal{N}$ (the families of meager and Lebesgue-measure-zero subsets of the reals; e.g. $\mathrm{cof}(\mathcal{N})$ is the least size of a cofinal subfamily of $\mathcal{N}$ under inclusion), these form van Douwen's diagram and Cichoń's diagram; e.g. $\mathfrak{b} \le \mathfrak{d} \le \mathrm{cof}(\mathcal{N})$ holds in ZFC. A persistent theme of the field is to locate each newly defined invariant in these diagrams.

Two “dualized” families of invariants arise by replacing subsets of $\omega$ by partitions of $\omega$ (the same move as in the dual Ramsey theorem of Carlson and Simpson, the statement that every finite coloring of the infinite partitions of $\omega$ is constant on all partitions coarser than some one partition). Write $(\omega)^\omega$ for the set of partitions of $\omega$ into infinitely many blocks, and view a partition as the set of pairs of naturals it merges. For $X, Y \in (\omega)^\omega$ let $Y \le^* X$ (“ $X$ is almost coarser than $Y$ ”) mean that every pair merged by $Y$ is merged by $X$ except finitely many, and let $Y \perp X$ ( $Y$ orthogonal to $X$ ) mean that only finitely many pairs of naturals are merged by both, i.e. their common refinement is almost the partition into singletons. Under the general lattice scheme of Majcher-Iwanow, the dual reaping number $\mathfrak{r}_d$ is the least size of a family $\mathcal{R} \subseteq (\omega)^\omega$ such that every $X \in (\omega)^\omega$ has some $Y \in \mathcal{R}$ with $Y \le^* X$ or $Y \perp X$ , and the dual splitting number $\mathfrak{s}_d$ is defined by the analogous splitting condition in the same scheme. These invariants were analyzed by Minami, who proved $\mathfrak{r}_d \le \mathrm{non}(\mathcal{M}), \mathrm{non}(\mathcal{N})$ , and by Kamo (in Minami's paper), who proved $\mathfrak{r}_d \le \mathfrak{d}$ and $\mathfrak{s}_d \ge \mathfrak{b}$ ; Majcher-Iwanow later recorded $\mathfrak{r}_d \le \min(\mathfrak{r}, \mathfrak{d}, \mathrm{non}(\mathcal{M}), \mathrm{non}(\mathcal{N}))$ and $\max(\mathrm{cov}(\mathcal{N}), \mathrm{cov}(\mathcal{M}), \mathfrak{s}, \mathfrak{b}) \le \mathfrak{s}_d$ . Minami also established the consistency of $\mathfrak{r}_d > \mathfrak{b}$ , $\mathfrak{s}_d < \mathfrak{d}$ , $\mathfrak{s}_d < \mathfrak{r}$ , $\mathfrak{r}_d < \mathrm{add}(\mathcal{M})$ and $\mathfrak{s}_d > \mathrm{cof}(\mathcal{M})$ , so the dual cardinals are genuinely off-diagonal.

To study these dual cardinals, Minami introduced pair-indexed variants. Call $\mathcal{A} \subseteq [\omega]^2$ unbounded if for every $k \in \omega$ some $a \in \mathcal{A}$ is disjoint from $k = \{0, \dots, k-1\}$ . A set $X \subseteq \omega$ pair-splits $\mathcal{A}$ if $|a \cap X| = 1$ for infinitely many $a \in \mathcal{A}$ (each such 2-element set crosses $X$ , meeting both $X$ and its complement). The pair-splitting number $\mathfrak{s}_{pair}$ is the least size of a family pair-splitting every unbounded $\mathcal{A}$ ; dually, the pair-reaping number $\mathfrak{r}_{pair}$ is the least size of a family of unbounded $\mathcal{A} \subseteq [\omega]^2$ no single $X$ pair-splits all of. Hrušák–Meza-Alcántara–Minami (HMM) proved $\mathfrak{s} \le \mathfrak{s}_{pair} \le \mathrm{non}(\mathcal{M}), \mathrm{non}(\mathcal{N})$ and $\mathrm{cov}(\mathcal{M}), \mathrm{cov}(\mathcal{N}) \le \mathfrak{r}_{pair} \le \mathfrak{r}, \mathfrak{s}_d$ , that $\mathfrak{s}_{pair} = \mathfrak{s}_n$ and $\mathfrak{r}_{pair} = \mathfrak{r}_n$ for the $n$ -element analogues with $n \ge 3$ , the consistency of $\mathfrak{s}_{pair} > \mathfrak{d}$ and of $\mathfrak{r}_{pair} < \mathfrak{b}$ , and $\mathfrak{r}_{pair} \le \mathfrak{l}$ and $\mathfrak{s}_{pair} \ge \mathrm{trans\text{-}add}(\mathcal{N})$ , where $\mathfrak{l}$ is Kada's slalom invariant (the least $\kappa$ such that for every $h \in \omega^\omega$ there are $\kappa$ many slaloms — functions $\varphi$ with $\varphi(n) \subseteq \omega$ and $|\varphi(n)| \le 2^n$ — that jointly capture every $f < h$ pointwise, in the sense that some slalom contains $f(n)$ for all but finitely many $n$ ) and $\mathrm{trans\text{-}add}(\mathcal{N})$ is Pawlikowski's transitive additivity of the null ideal, i.e. the least size of a $\le^*$ -bounded family $\mathcal{F} \subseteq \omega^\omega$ such that every slalom $\varphi$ (with $|\varphi(n)| \le 2^n$ ) fails to contain $f(n)$ for infinitely many $n$ , for some $f \in \mathcal{F}$ . Recently Weinert proved that the unbounded evasion number $\mathfrak{e}_{ubd}$ (the least size of a family of functions from $\prod_n f(n)$ , for some unbounded $f$ , that no single predictor — a function assigning finitely many candidate values to each coordinate — predicts on all but finitely many coordinates) satisfies $\mathfrak{e}_{ubd} \le \mathfrak{s}_{pair}$ ; Shelah–Steprāns generalized pair-splitting from pairs to arbitrary finite sets.

The same paper connects these invariants to ideals on $\omega$ . An ideal is a family $\mathcal{I} \subseteq \mathcal{P}(\omega)$ closed under finite unions and subsets and containing all finite sets; it is tall if every infinite set contains an infinite member, and Borel or $F_\sigma$ according to its complexity as a subset of the Polish space $2^\omega \cong \mathcal{P}(\omega)$ . For a tall ideal $\mathcal{I}$ , the star-uniformity $\mathrm{non}^*(\mathcal{I})$ is the least size of $\mathcal{H} \subseteq [\omega]^\omega$ such that every $I \in \mathcal{I}$ has finite intersection with some $H \in \mathcal{H}$ ; dually, the star-covering $\mathrm{cov}^*(\mathcal{I})$ is the least size of $\mathcal{A} \subseteq \mathcal{I}$ such that every infinite $X$ has infinite intersection with some $A \in \mathcal{A}$ . Ideals are compared by the Katětov order $\mathcal{I} \le_K \mathcal{J}$ ( $f^{-1}[I] \in \mathcal{J}$ for all $I \in \mathcal{I}$ , some function $f$ ) and its Katětov-Blass refinement with $f$ finite-to-one. HMM's main structural result is $\mathfrak{s}_{pair} = \mathrm{cov}^*(\mathcal{G}_{FC})$ and $\mathfrak{r}_{pair} \le \mathrm{non}^*(\mathcal{G}_{FC})$ with $\mathrm{cov}(\mathcal{N}) \le \mathrm{non}^*(\mathcal{G}_{FC})$ , where $\mathcal{G}_{FC}$ is the $F_\sigma$ ideal of finitely chromatic graphs ( $A \subseteq [\omega]^2$ such that the graph $(\omega, A)$ has finite chromatic number); they also answered a question of Solecki by showing $\mathcal{G}_{FC}$ is not locally Katětov-minimal among ideals failing Fatou's lemma. For ideal invariants they proved (Corollary 4.6) that every $F_\sigma$ ideal $\mathcal{I}$ satisfies $\mathrm{non}^*(\mathcal{I}) \le \mathfrak{l}$ and $\mathrm{cov}^*(\mathcal{I}) \ge \mathrm{trans\text{-}add}(\mathcal{N})$ . The Hrušák 2011 survey records the dichotomy that every Borel ideal $\mathcal{I}$ has $\mathrm{non}^*(\mathcal{I}) = \omega$ or $\mathcal{ED}_{fin} \le_{KB} \mathcal{I}$ ( $\mathcal{ED}_{fin}$ being the restriction to the triangle $\{(m,n) : n \le m\}$ of the ideal $\mathcal{ED}$ of sets $A \subseteq \omega \times \omega$ whose vertical sections are eventually uniformly bounded in size), which reduces much of the general Borel theory to the eventually-different test case.

Question 4.8 of the source paper asks exactly two things and is the object of this problem: (1) whether $\mathfrak{r}_d \le \mathfrak{s}_{pair}$ in ZFC, and (2) whether $\mathrm{non}^*(\mathcal{I}) \le \mathrm{cof}(\mathcal{N})$ for every Borel ideal $\mathcal{I}$ . The paper's introduction states explicitly that “it is not known whether $\mathfrak{r}_d \le \mathfrak{s}_{pair}$ or not”. Both questions sit at the boundary between the partition-dual cardinals, the pair-indexed cardinals, and the ideal-theoretic star-invariants, and neither the source nor any later work answers either one.

Problem Statement

The problem is Question 4.8 of Hrušák–Meza-Alcántara–Minami (J. Symbolic Logic 75 (2010), 661–677), stated verbatim as two parts; the common objective is to fix, in ZFC, the position of the pair-indexed and partition-dual cardinal invariants introduced in that paper relative to the classical invariants of Cichoń's diagram.

Part (1): Is $\mathfrak{r}_d \le \mathfrak{s}_{pair}$ ? Here $\mathfrak{s}_{pair}$ is the pair-splitting number: the least cardinality of a family $\mathcal{S} \subseteq [\omega]^\omega$ such that for every unbounded $\mathcal{A} \subseteq [\omega]^2$ (unbounded meaning: for every $k \in \omega$ some $a \in \mathcal{A}$ satisfies $a \cap k = \emptyset$ ) there is $X \in \mathcal{S}$ with $|a \cap X| = 1$ for infinitely many $a \in \mathcal{A}$ . And $\mathfrak{r}_d$ is the dual reaping number: writing $(\omega)^\omega$ for the set of partitions of $\omega$ into infinitely many blocks and viewing a partition as the set of pairs of naturals it merges, $\mathfrak{r}_d$ is the least cardinality of a family $\mathcal{R} \subseteq (\omega)^\omega$ such that for every $X \in (\omega)^\omega$ there is $Y \in \mathcal{R}$ with $Y \le^* X$ or $Y \perp X$ , where $Y \le^* X$ means that every pair merged by $Y$ is merged by $X$ except finitely many ( $X$ is almost coarser than $Y$ ), and $Y \perp X$ means that only finitely many pairs of naturals are merged by both.

Part (2): If $\mathcal{I}$ is a Borel ideal on $\omega$ , does $\mathrm{non}^*(\mathcal{I}) \le \mathrm{cof}(\mathcal{N})$ hold in ZFC? Here $\mathrm{non}^*(\mathcal{I})$ is the star-uniformity of $\mathcal{I}$ (the least size of $\mathcal{H} \subseteq [\omega]^\omega$ such that every $I \in \mathcal{I}$ has finite intersection with some $H \in \mathcal{H}$ ; non-tall ideals have countable $\mathrm{non}^*$ , so the question concerns tall Borel ideals), and $\mathrm{cof}(\mathcal{N})$ is the cofinality of the Lebesgue-null ideal (the least size of a cofinal subfamily of $\mathcal{N}$ under inclusion).

No additional hypotheses (e.g. Martin's Axiom, cardinal-characteristic arithmetic beyond ZFC) may be assumed or imposed.

References

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