Open Problems in Mathematical Logic — Problem 26

Definitions and mathematical statement

The supplied record refers to a family of games $G_k$, but does not define their rules, legal plays, information available to the players, or winning conditions. It also does not explicitly specify the domain of the index $k$.

Let $I$ denote the intended, unspecified index domain. For a fully specified game $G_k$, let $\Sigma_1(G_k)$ denote its set of legal strategies for Player 1, and let $\operatorname{Win}_1(G_k,\sigma)$ mean that $\sigma$ is a winning strategy for Player 1 under that game's rules. These are placeholders for definitions absent from the record.

The question can therefore be recorded only schematically as follows: is it true that \[ \forall k\in I\;\left(k\geq 4\Longrightarrow \exists\sigma\in\Sigma_1(G_k)\;\operatorname{Win}_1(G_k,\sigma)\right)? \] The source reports that Player 1 has a winning strategy for $k=2,3$, attributing this result to [Ba], and describes the cases $k\geq4$ as unknown. Neither this reported result nor the supplied open status is independently verified here. A precise mathematical statement requires the missing definition of $G_k$ and its index domain.

Short English statement

For each intended index $k\geq4$, does Player 1 have a winning strategy in $G_k$? The game and index domain are not defined in the supplied record, so this remains a schematic statement requiring mathematical review.

Sources

• Local UnsolvedMath registry: permanent ID $801014$, source problem code catalogue entry, titled Open Problems in Mathematical Logic --- Problem 26. The supplied registry labels the problem open and attributes it to Barmpalias, Gao et al., with proposed year 2026.
• Supplied source list: Barmpalias, Gao et al., Open Problems in Mathematical Logic (2026), Problem 26, https://arxiv.org/abs/2608.26628. The record gives extraction method arxiv_latex and access date 2026-10-16. These provenance details are supplied metadata; no source visit or independent verification is claimed.
• [Ba]: the supplied statement attributes the winning-strategy result for $k=2,3$ to this reference label. No corresponding author, title, publication details, or link is supplied, so this reference cannot be resolved from the record.