Selection Principles Milestone Problems — Problem 39 ([PolarPar, Problem 1])

Definitions and mathematical statement

The supplied record asks about a polarized partition relation for open $\omega$-covers. Its excerpt does not specify the ambient class of spaces, the convention on countability of covers, or the quantification of $k$. The following gives a precise parameterized formulation under the Cartesian-product interpretation of the displayed relation; these missing conventions require confirmation from the source.

Let $\mathscr X$ be a specified class of topological spaces. For each $X\in\mathscr X$, let $\mathscr U_X$ specify the admissible open covers of $X$. Define \[ \Omega(X)=\left\{\mathcal U\in\mathscr U_X:X\notin\mathcal U\ \text{and}\ (\forall F\subseteq X\text{ finite})(\exists U\in\mathcal U)\ F\subseteq U\right\}. \] Thus every finite subset of $X$ is contained in one member of an $\omega$-cover, and no member is the entire space. Whether $\mathscr U_X$ consists of all open covers or only countable open covers is left unresolved by the supplied excerpt.

The property $\mathsf S_1(\Omega,\Omega)$ means \[ (\forall (\mathcal U_n)_{n\geq1}\in\Omega(X)^{\{1,2,\ldots\}}) (\exists (U_n)_{n\geq1})\quad (\forall n\geq1)\ U_n\in\mathcal U_n \quad\text{and}\quad \{U_n:n\geq1\}\in\Omega(X). \] For a positive integer $k$, write $P_k(X)$ for the following Cartesian-product reading of \[ \binom{\Omega}{\Omega}\longrightarrow \left[\begin{matrix}\Omega\\\Omega\end{matrix}\right]_{k/<3}: \] for every $\mathcal U,\mathcal V\in\Omega(X)$ and every coloring $c:\mathcal U\times\mathcal V\to\{1,\ldots,k\}$, there are subfamilies $\mathcal U'\subseteq\mathcal U$ and $\mathcal V'\subseteq\mathcal V$ such that \[ \mathcal U',\mathcal V'\in\Omega(X) \quad\text{and}\quad \left|c[\mathcal U'\times\mathcal V']\right|<3. \] In particular, the retained rectangle uses at most two colors. This definition treats the coloring as defined on the entire ordered Cartesian product; the excerpt supplies no definition confirming this convention or specifying additional restrictions.

Let $P$ denote the property intended by the source notation, once its conventions have been fixed. The first question is whether \[ (\forall X\in\mathscr X)\qquad P(X)\ \Longleftrightarrow\ X\models\mathsf S_1(\Omega,\Omega). \] The excerpt does not determine whether $P=P_k$ for a fixed $k$, or whether $P(X)$ means $(\forall k\geq1)P_k(X)$. For $k=1,2$, the displayed Cartesian-product condition is automatic, so this distinction is material.

For a property $Q$, define its critical cardinality relative to $\mathscr X$, when the minimum exists, by \[ \operatorname{non}_{\mathscr X}(Q) =\min\{|X|:X\in\mathscr X\text{ and }X\text{ fails }Q\}. \] Let $\mathcal M$ be the ideal of meager subsets of ${\mathbb{R}}$, and define \[ \mathsf{cov}(\mathcal M) =\min\left\{|\mathcal A|:\mathcal A\subseteq\mathcal M, \ \bigcup\mathcal A={\mathbb{R}}\right\}. \] If the proposed equivalence fails, the second question is whether \[ \operatorname{non}_{\mathscr X}(P) =\operatorname{non}_{\mathscr X}\left(\mathsf S_1(\Omega,\Omega)\right) =\mathsf{cov}(\mathcal M). \] The identification of the critical cardinality of $\mathsf S_1(\Omega,\Omega)$ with $\mathsf{cov}(\mathcal M)$ is supplied parenthetically by the source. It is recorded as source context, subject to the source's missing ambient conventions, rather than asserted here for an arbitrary $\mathscr X$.

Short English statement

The source asks whether a polarized coloring property for open $\omega$-covers is equivalent to selecting one member from each of a sequence of $\omega$-covers so that the selected family is again an $\omega$-cover. If equivalence fails, it asks whether the least size of a counterexample is nevertheless $\mathsf{cov}(\mathcal M)$. The ambient spaces, cover convention, and quantification of $k$ require clarification.

Sources

• Local UnsolvedMath registry, permanent ID $800666$, exact source problem code catalogue entry; collection: AMR Open Problem Lists. The supplied record labels the problem open; this is a supplied status, not an independently verified current assessment.
• Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 39. Supplied source URL: https://arxiv.org/abs/math/0312182. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of access or verification for this draft.
• The source gives the attribution/context label [PolarPar, Problem 1]. No full bibliographic entry for PolarPar is included in the supplied record. The registry attributes the problem to Tsaban and records the year 2003; the underlying attribution cannot be resolved from the excerpt alone.