Structure of Extremal Product-Free Subsets

Definitions and mathematical statement

For a positive integer $n$, let $A_n$ be the group of even permutations of $\{1,\ldots,n\}$, with multiplication given by composition. A subset $A\subseteq A_n$ is product-free if \[ AA\cap A=\varnothing,\qquad AA=\{xy:x,y\in A\}. \] Thus there are no $x,y,z\in A$ with $xy=z$; the elements in this condition need not be distinct. Define \[ m(n)=\max\{|B|:B\subseteq A_n,\ BB\cap B=\varnothing\}, \] and the family of extremal sets by \[ \mathcal E_n=\{A\subseteq A_n:AA\cap A=\varnothing,\ |A|=m(n)\}. \] The supplied problem asks for a structural characterization of $\mathcal E_n$ and of product-free subsets whose cardinalities are close to $m(n)$. One can parameterize cardinality deficits by \[ \mathcal F_n(t)=\{A\subseteq A_n:AA\cap A=\varnothing,\ |A|\ge m(n)-t\}, \qquad t\in\{0,\ldots,m(n)\}. \] This parameterization makes the size deficit explicit, but the source does not specify the intended range or dependence of $t$ on $n$, a notion of structural closeness, or quantitative stability bounds. It also gives no explicit restriction on $n$. Consequently the near-extremal request remains a qualitative research question rather than a uniquely specified quantified stability assertion.

The supplied curation labels the problem partially solved and reports that a cited theorem supplies extremal structure and stability for sufficiently large $n$, while the unrestricted structure question is not fully resolved for every $n$. No threshold or precise theorem statement is supplied here; this assessment is retained as source information, not as an independently verified mathematical claim.

Short English statement

Describe the maximum-size product-free subsets of the alternating group $A_n$, and describe product-free subsets with sizes close to that maximum. The source leaves the meaning and quantitative scope of near-extremality unspecified. Its dated curation reports results for sufficiently large $n$ and labels the broader question partially solved.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30005103$, title Structure of Extremal Product-Free Subsets, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The proposed year is recorded as 2022; no individual proposer is supplied.
• Supplied report citation: Combinatorics, Probability and Computing (2023), Oberwolfach Reports, pp. 1165--1237, DOI 10.4171/owr/2022/22. https://doi.org/10.4171/owr/2022/22. The original extracted question is "What can be said about the structure of A?" The supplied clean statement identifies the target as extremal and near-extremal product-free subsets of $A_n$.
• Supplied literature reference: Keevash--Lifshitz--Minzer, product-free alternating groups. https://doi.org/10.1007/s00222-024-01273-1.
• Supplied literature reference, labeled primary preprint: https://arxiv.org/abs/2205.15191.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. It labels the problem partially solved and reports extremal structure and stability for sufficiently large $n$, with the vague unrestricted structure question not fully resolved for every $n$. The source explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and records that the official report was checked directly. These are supplied verification claims; no independent source inspection is asserted in this draft.