Selection Principles Milestone Problems — Problem 33 ([coc5, Problem 3])

Definitions and mathematical statement

The supplied record asks about two selection principles involving $\mathcal D_\Omega$, but does not define this family or specify the ambient objects to which the properties apply. Consequently, the following gives the standard abstract selection and game formulations; identifying their intended parameters remains necessary for a complete mathematical statement.

Let $U$ be a set and let $\mathscr A\subseteq\mathcal P(U)\setminus\{\varnothing\}$. These are auxiliary parameters, not an asserted definition of $\mathcal D_\Omega$. Define \[ \mathsf S_1(\mathscr A,\mathscr A) \quad\Longleftrightarrow\quad \forall (A_n)_{n\in{\mathbb{N}}}\in\mathscr A^{{\mathbb{N}}}\ \exists (a_n)_{n\in{\mathbb{N}}}\in U^{{\mathbb{N}}}\ \left[(\forall n\in{\mathbb{N}}\;a_n\in A_n)\ \land\ \{a_n:n\in{\mathbb{N}}\}\in\mathscr A\right]. \] Define \[ \mathsf S_{\mathrm{fin}}(\mathscr A,\mathscr A) \quad\Longleftrightarrow\quad \forall (A_n)_{n\in{\mathbb{N}}}\in\mathscr A^{{\mathbb{N}}}\ \exists (F_n)_{n\in{\mathbb{N}}}\ \left[(\forall n\in{\mathbb{N}}\;F_n\subseteq A_n\text{ is finite})\ \land\ \bigcup_{n\in{\mathbb{N}}}F_n\in\mathscr A\right]. \] Here ${\mathbb{N}}=\{0,1,2,\ldots\}$, and finite selections are allowed to be empty in this template.

In the corresponding game $\mathsf G_1(\mathscr A,\mathscr A)$, inning $n$ consists of ONE choosing $A_n\in\mathscr A$, followed by TWO choosing $a_n\in A_n$. TWO wins exactly when $\{a_n:n\in{\mathbb{N}}\}\in\mathscr A$; otherwise ONE wins. In $\mathsf G_{\mathrm{fin}}(\mathscr A,\mathscr A)$, TWO instead chooses a finite subset $F_n\subseteq A_n$, and wins exactly when $\bigcup_{n\in{\mathbb{N}}}F_n\in\mathscr A$. A strategy for ONE specifies a legal move at each finite legal history at which ONE moves, and is winning if every infinite legal play following it is won by ONE.

With $\mathscr A$ replaced by the intended, presently unspecified $\mathcal D_\Omega$, the two questions are whether \[ \mathsf S_1(\mathcal D_\Omega,\mathcal D_\Omega) \quad\overset{?}{\Longleftrightarrow}\quad \text{ONE has no winning strategy in }\mathsf G_1(\mathcal D_\Omega,\mathcal D_\Omega) \] and whether \[ \mathsf S_{\mathrm{fin}}(\mathcal D_\Omega,\mathcal D_\Omega) \quad\overset{?}{\Longleftrightarrow}\quad \text{ONE has no winning strategy in }\mathsf G_{\mathrm{fin}}(\mathcal D_\Omega,\mathcal D_\Omega). \] These are separate proposed equivalences. Their intended scope cannot be fixed from the supplied record: the definition of $\mathcal D_\Omega$, the class of ambient objects, and any additional hypotheses are missing. The game conventions above are the standard abstract template and require confirmation against the cited source; they are not recovered source definitions.

Short English statement

For each of two selection properties involving $\mathcal D_\Omega$, ask whether the property is equivalent to ONE having no winning strategy in its corresponding infinite selection game. The supplied record omits the definition of $\mathcal D_\Omega$ and the ambient hypotheses, so this draft gives a conditional formal template requiring human review.

Sources

• UnsolvedMath local registry, permanent ID $800660$, exact source problem code catalogue entry; supplied title: "Selection Principles Milestone Problems --- Problem 33 ([coc5, Problem 3])." The registry attributes the record to Tsaban, with proposed year 2003, and labels it open. This status is supplied metadata, not an independently verified mathematical or literature assessment.
• Supplied source list: Tsaban, Selection Principles in Mathematics: A Milestone of Open Problems (2003), Problem 33. Supplied 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 independent access or verification.
• Supplied attribution/context: [coc5, Problem 3]. The bibliographic identity of coc5 is not included in the supplied record, so the underlying reference and its precise hypotheses remain unresolved.