Open boundary cases of the negative strong polarized partition relation: the sigma-reaping number under $\mathfrak{r}_\sigma=\mathfrak{c}$ and the real-valued measurable endpoint

The strong polarized partition relation $\binom{\kappa}{\omega}\rightarrow\binom{\kappa}{\omega}^{1,1}_2$ (every 2-coloring of $\kappa\times\omega$ has a monochromatic rectangle $A\times B$ with $|A|=\kappa$ and $B$ infinite) was mapped by Garti and Shelah (2018) against the splitting number $\mathfrak{s}$, the reaping number $\mathfrak{r}$, and real-valued measurable cardinals, leaving four boundary cases open. Chen, Garti and Weinert (2020) later settled the singular-cofinality consistency questions (Question 1.6 and the $\kappa=\mathfrak{c}$ instance of Question 1.8(alpha)), producing a model with $\mathfrak{r}<\mathfrak{c}$, uncountable cofinality of $\mathfrak{c}$, and the negative relation. The two surviving cases are: (a) whether $\mathfrak{r}_\sigma=\mathfrak{c}$ provably implies $\binom{\mathfrak{r}_\sigma}{\omega}\nrightarrow\binom{\mathfrak{r}_\sigma}{\omega}^{1,1}_2$, where $\mathfrak{r}_\sigma$ is Vojtáš's sigma-reaping number (Question 1.8(beta) of the source); and (b) whether the negative relation $\binom{\theta}{\omega}\nrightarrow\binom{\aleph_1}{\omega}^{1,1}_2$, proved from a real-valued measurable $\kappa$ for all $\theta<\kappa$ via a Sierpiński-set construction, extends to the endpoint $\theta=\kappa$. Both remain open: no later work connects sigma-reaping to polarized relations, and the measure-theoretic construction is guaranteed only strictly below the measurable cardinal. Part (a) is entangled with Vojtáš's 1992 problem of separating $\mathfrak{r}$ from $\mathfrak{r}_\sigma$; part (b) with the consistency of the positive relation at a real-valued measurable cardinal.

Background

In the partition calculus of Erdős, Hajnal and Rado, the strong polarized relation $\binom{\kappa}{\lambda}\rightarrow\binom{\kappa}{\lambda}^{1,1}_2$ asserts that for every coloring $c:\kappa\times\lambda\rightarrow 2$ there are sets $A\subseteq\kappa$, $B\subseteq\lambda$ with $|A|=\kappa$, $|B|=\lambda$ such that $c$ is constant on the rectangle $A\times B$; the negation is written $\nrightarrow$, and unbalanced variants such as $\binom{\kappa}{\omega}\rightarrow\binom{\aleph_1}{\omega}^{1,1}_2$ ask only for $|A|=\aleph_1$. Throughout, $\mathfrak{c}=2^{\aleph_0}$ is the cardinality of the continuum.

The boundary behavior of this relation is governed by two cardinal characteristics of the continuum. A set $S\subseteq\omega$ splits $B\in[\omega]^\omega$ if both $S\cap B$ and $B\setminus S$ are infinite. The splitting number $\mathfrak{s}$ is the least size of a family of subsets of $\omega$ splitting every infinite $B\subseteq\omega$; the reaping (or unsplitting) number $\mathfrak{r}$ is the least size of a family $\mathcal{R}\subseteq[\omega]^\omega$ such that no single set splits all members of $\mathcal{R}$. Always $\aleph_1\le\mathfrak{s},\mathfrak{r}\le\mathfrak{c}$. Vojtáš introduced the sigma-reaping number $\mathfrak{r}_\sigma$: the least size of a family $\mathcal{R}\subseteq[\omega]^\omega$ which cannot be split by countably many sets, i.e. for every countable family $\{X_n:n<\omega\}\subseteq[\omega]^\omega$ some $A\in\mathcal{R}$ is not split by any $X_n$ (equivalently, for each $n$ either $A\setminus X_n$ or $A\cap X_n$ is finite). Clearly $\mathfrak{r}\le\mathfrak{r}_\sigma\le\mathfrak{c}$, and whether $\mathfrak{r}<\mathfrak{r}_\sigma$ is consistent has been open since 1992.

Garti and Shelah (2018) mapped the strong polarized relation against these invariants. In ZFC: below the splitting number the relation is governed by cofinality (for $\kappa<\mathfrak{s}$, the positive relation holds iff $\mathrm{cf}(\kappa)>\aleph_0$); above the reaping number, $\mathfrak{r}<\kappa\le\mathfrak{c}$ with $\mathfrak{r}<\mathrm{cf}(\kappa)$ gives the positive relation. Under $\mathfrak{r}=\mathfrak{c}$ the picture flips: their Main Claim proves $\binom{\kappa}{\omega}\nrightarrow\binom{\kappa}{\omega}^{1,1}_2$ for every $\kappa\in[\mathrm{cf}(\mathfrak{c}),\mathfrak{c}]$, and Theorem 1.7 shows that already $\mathfrak{r}_\sigma=\mathfrak{c}$ forces $\mathfrak{s}\le\mathrm{cf}(\mathfrak{c})$.

A real-valued measurable cardinal is an uncountable cardinal $\kappa$ carrying a countably additive probability measure on $\mathcal{P}(\kappa)$, vanishing on singletons, that is $\kappa$-additive (a union of fewer than $\kappa$ null sets is null). If $\kappa>\mathfrak{c}$ such a measure is necessarily two-valued and $\kappa$ is a measurable cardinal in the usual sense, so the case of interest here is $\kappa\le\mathfrak{c}$; in that range such a $\kappa$ is weakly inaccessible, hence regular, and by a theorem of Solovay the existence of a real-valued measurable cardinal is equiconsistent with the existence of a (two-valued) measurable cardinal. A Sierpiński set is a set of reals meeting every Lebesgue-null set in at most countably many points; from a $\kappa$-additive measure on $\kappa$ one obtains Sierpiński sets of every cardinality $\theta<\kappa$ (Fremlin). This underlies Theorem 2.3 of the source paper: if $\kappa$ is real-valued measurable then $\binom{\theta}{\omega}\nrightarrow\binom{\aleph_1}{\omega}^{1,1}_2$ for every $\theta<\kappa$. The proof breaks exactly at the endpoint $\theta=\kappa$, where the Sierpiński-set extraction is no longer guaranteed. Two further results shape the endpoint: the existence of a real-valued measurable cardinal implies $\mathfrak{s}=\aleph_1$ (Corollary 2.4), and $\mathfrak{r}\ge\kappa$ (Claim 2.5), so when $\kappa=\mathfrak{c}$ is real-valued measurable the balanced negative relation at $\kappa$ follows; the unbalanced endpoint question of part (b) remains open at every real-valued measurable $\kappa$, $\kappa=\mathfrak{c}$ included. Since a monochromatic rectangle with $A\in[\kappa]^\kappa$ can always be thinned to $A\in[\kappa]^{\aleph_1}$, the balanced positive relation at such a $\kappa$ implies the unbalanced one; the balanced form is Question 0.7 of the source paper (Problem 3.12 of Garti and Shelah 2016).

Chen, Garti and Weinert subsequently resolved the singular-cardinal cofinality questions left open by this work (Questions 1.6 and 1.8(alpha) of the source paper, the latter at $\kappa=\mathfrak{c}$): forcing the independence number $\mathfrak{i}$ (the least size of a maximal independent family of subsets of $\omega$, where independent means every finite Boolean combination of members is infinite) to be $\aleph_1$ while $\mathfrak{c}=\aleph_{\omega_1}$ yields $\mathfrak{r}=\aleph_1<\mathfrak{c}$, $\mathrm{cf}(\mathfrak{c})=\omega_1>\aleph_0$, and the negative relation $\binom{\mathfrak{c}}{\omega}\nrightarrow\binom{\mathfrak{c}}{\omega}^{1,1}_2$, via their result that $\mathfrak{b}=\mathfrak{d}$ (the unbounding and dominating numbers for $\omega^\omega$ under eventual domination) implies a negative polarized square-bracket relation, plus a cofinality-lifting claim. What survives untouched are the two boundary cases stated below.

Problem Statement

Determine the status of the strong polarized partition relation at the two boundary points of the reaping-number framework left open by Garti and Shelah (2018), after the Chen-Garti-Weinert (2020) resolution of the singular cofinality questions. The common objective is to complete the classification of the negative relation $\binom{\kappa}{\omega}\nrightarrow\binom{\kappa}{\omega}^{1,1}_2$ (and its unbalanced variant) at the distinguished cardinals $\kappa\le\mathfrak{c}$ where the splitting/reaping and measure-theoretic methods of the source paper stall; each part is settled by a ZFC derivation or by a relative-consistency construction.

(a) The sigma-reaping boundary (Question 1.8(beta) of the source paper). Assume $\mathfrak{r}_\sigma=\mathfrak{c}$. Is it provable in ZFC that $\binom{\mathfrak{r}_\sigma}{\omega}\nrightarrow\binom{\mathfrak{r}_\sigma}{\omega}^{1,1}_2\ ?$ Either give a ZFC proof, or produce a model of ZFC with $\mathfrak{r}_\sigma=\mathfrak{c}$ in which the positive relation $\binom{\mathfrak{r}_\sigma}{\omega}\rightarrow\binom{\mathfrak{r}_\sigma}{\omega}^{1,1}_2$ holds.

(b) The real-valued measurable endpoint (the $\theta=\kappa$ gap in Theorem 2.3 of the source paper, together with its balanced counterpart, Question 0.7 there). Let $\kappa$ be a real-valued measurable cardinal with $\kappa\le\mathfrak{c}$. Does the negative relation of Theorem 2.3 extend to the endpoint, i.e. is $\binom{\kappa}{\omega}\nrightarrow\binom{\aleph_1}{\omega}^{1,1}_2$ provable in ZFC? The source's proof obtains the relation for every $\theta<\kappa$ from a Sierpiński set carried by the $\kappa$-additive measure, and it is unknown whether such a construction (or any other) works at $\theta=\kappa$. A refutation consists of a model of ZFC with a real-valued measurable $\kappa\le\mathfrak{c}$ in which the positive relation $\binom{\kappa}{\omega}\rightarrow\binom{\aleph_1}{\omega}^{1,1}_2$ holds.

References

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