Equal Products from Disjoint Subsets of Finite Groups

Definitions and mathematical statement

Let $k\geq 2$ be an integer, let $G$ be a finite group of order $n=|G|$, and let $S\subseteq G$. An ordering of a subset $T\subseteq S$ is a list $(t_1,\ldots,t_m)$ containing each element of $T$ exactly once, where $m=|T|$. Its ordered product is $t_1\cdots t_m$; in a noncommutative group this product may depend on the ordering. The empty ordered product is conventionally $1_G$.

The supplied question asks whether, for every integer $k\geq 2$, there exists a constant $c(k)>0$, depending only on $k$, such that for every finite group $G$ and every $S\subseteq G$, \[ |S|\geq c(k)\log |G| \quad\Longrightarrow\quad \begin{gathered} \text{there exist }T_1,\ldots,T_k\subseteq S \text{ with }T_i\cap T_j=\varnothing\quad(i\neq j),\\ \text{orderings }(t_{i,1},\ldots,t_{i,m_i})\text{ of }T_i, \text{ and }g\in G\text{ such that}\\ t_{i,1}\cdots t_{i,m_i}=g\qquad(1\leq i\leq k). \end{gathered} \] Here $\log$ may be taken to be the natural logarithm, since changing to any fixed base greater than $1$ only rescales the constant.

Unresolved formulation issue. The supplied statement does not require the subsets $T_i$ to be nonempty or distinct. Under the conventional empty-product definition, taking every $T_i=\varnothing$ satisfies the displayed conclusion for every $S$, making the literal assertion trivial. Requiring each $m_i\geq 1$ would give a different, substantive question, but that requirement is absent from the supplied wording and is not adopted here as a verified correction. Moreover, with such a nonemptiness requirement, the stated quantification over all finite groups fails for $|G|=1$: the choice $S=\varnothing$ satisfies the size hypothesis because $\log 1=0$. An additional boundary condition or revised threshold would therefore also need clarification.

Short English statement

The source asks whether a subset of a finite group whose size is at least a constant times the logarithm of the group order contains $k$ pairwise disjoint subsets with equal products after suitable ordering. The wording permits empty subsets, making the literal question trivial; a substantive interpretation requires clarification of nonemptiness and the group of order $1$.

Sources

• Local UnsolvedMath registry, permanent ID $30005355$, title Equal Products from Disjoint Subsets of Finite Groups, source problem code catalogue entry, in the supplied Oberwolfach Reports Open Problems collection. The record lists proposed year 2023 and supplies no named proposer.
• Supplied report citation: Combinatorics (2023), Oberwolfach Reports, DOI 10.4171/owr/2023/1, pp. 5--89. https://doi.org/10.4171/owr/2023/1. The record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Alweiss--Lovett--Wu--Zhang, Improved Sunflower Bounds. https://doi.org/10.4007/annals.2021.194.3.5.
• Supplied dated literature curation, checked 2026-08-22: the record labels the problem open and states that the requested logarithmic threshold is not known even for general abelian groups, while sunflower bounds give an additional log-logarithmic loss. This is the supplied dated assessment, not an independently verified literature conclusion; the source itself requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and reports a check against the official Oberwolfach report. That verification claim is source metadata and has not been independently verified in this draft.