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Claim. Theorem 6, as numbered in the authors' manuscript described on the source card, of F. Akman and P. A. Sissokho, Transversal coset partitions of groups, Beitr. Algebra Geom. 66 (2025), no. 2, 417--441, published online 2024-04-25, states: "Let $G$ be any group and $H_1,\ldots,H_r$ be distinct proper subgroups of $G$, with $2\leq r\leq 7$. Then any $\{H_1,\ldots,H_r\}$-transversal coset partition of $G$ must contain at least two cosets whose subgroups have the same index." A transversal partition uses at least one coset of each $H_i$. The cases $r\le4$ follow from the unit-fraction identity and the fact that two coprime indices cannot occur in a coset partition; for $5\le r\le7$ the authors report a computer enumeration of lists of distinct indices with reciprocal sum $1$, each of which contains $2$ or a coprime pair.

Covers. The case of Problem 274 with at most seven cosets: in a partition with pairwise different indices the subgroups are distinct, so no group has an exact covering by two to seven cosets of pairwise different sizes. Itabe's pending claim would raise the bound to seventeen cells.

Depends on. Nothing in this wiki; the theorem is the paper's own.

Acceptance. Refereed: Beiträge zur Algebra und Geometrie 66 (2025), no. 2, 417--441, doi:10.1007/s13366-024-00748-9, published online 2024-04-25, which gives the page its date. Not reviewed: the site's commentary does not mention the paper, and the site labels the problem OPEN. Not formalized: no Lean proof of the theorem is recorded.