Large Triple-Product-Free Subsets of Finite Groups

Definitions and mathematical statement

Do there exist an infinite family of finite groups $G_n$ with $|G_n|=n$ and subsets $S_n\subseteq G_n$ such that every solution of \[ ab^{-1}cd^{-1}ef^{-1}=1, \qquad a,b,c,d,e,f\in S_n, \] is trivial in the source's intended sense, and \[ |S_n|\gg n^{1/3}\ ? \] The product order is exact; the groups are not assumed abelian, so factors cannot be rearranged. Infinitely many groups is read as a genuine family with unbounded orders, not repeated isomorphic copies.

The term trivial solution is not defined. Assignments with $a=b$, $c=d$, and $e=f$ certainly give formal cancellation, but the source does not say whether these alone, all free-word cancellation patterns, equality of positive/negative multisets in an abelian convention, or some other class is intended. These choices give different restrictions. No trivial-solution definition is silently chosen.

There is also a size-quantifier ambiguity. With the standard Vinogradov meaning, $\gg$ would require a constant $c>0$ independent of the family index and an infinite set of orders with $|S_n|\ge c n^{1/3}$. If the intended question is growth genuinely beyond the cube-root scale, it might instead require $|S_n|/n^{1/3}\to\infty$ along the family. The assessment's phrase greater than $n^{1/3}$ does not decide this. Neither a divergent ratio nor a fixed power improvement $n^{1/3+\varepsilon}$ is silently substituted for the printed symbol.

The request is for some infinite collection of group orders, not one construction for every positive integer $n$. The source specifies no group class, prescribed construction, or further restrictions on the subsets. A precise extremal problem therefore requires both the forbidden nontrivial patterns and the intended meaning of the size comparison to be recovered.

Short English statement

The question seeks an infinite family of finite groups with large subsets avoiding nontrivial solutions of the exact six-factor word equation. The crucial notions of trivial solution and growth beyond or comparable to the cube-root scale are undefined.

Sources

• Local UnsolvedMath registry, permanent ID 30000902, code catalogue entry, "Large Triple-Product-Free Subsets of Finite Groups." Proposed year 2008; proposer unspecified.
• Combinatorics (2008), Oberwolfach Reports, pp. 5--78, DOI 10.4171/owr/2008/01; https://doi.org/10.4171/owr/2008/01. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied note says official-report content was checked and notation, context, and question form normalized without changing the source claim.
• Literature triage checked 2026-08-21: open. It says no later primary result in the supplied search gives the required infinite family at size greater than $n^{1/3}$. That wording does not specify the exact $\gg$ comparison or trivial-solution condition. Created 2026-08-25; updated 2026-09-16. Exact formulation and current status require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.