Largest product-free sets in finite groups
If every nontrivial representation of G has dimension at least D, is a largest product-free subset of G always of a simple combinatorial form coming from an action on a set of size D^{O(1)}?
Cite
@misc{cairn-product-free-sets-in-groups,
title = {Largest product-free sets in finite groups},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/product-free-sets-in-groups}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
} Also: CITATION.cff · Atom feed of results
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The problem
The question
A subset A of a group G is product-free if xy ∉ A for all x, y ∈ A. Babai and Sós asked how large a product-free set can be; Gowers showed that if every nontrivial irreducible representation of G has dimension at least D, then product-free sets have density at most about D^{-1/3}.
Natural large product-free sets come from actions: if G acts on a set X, x ∈ X and I ⊆ X, the set {g ∈ G : g(x) ∈ I and g(I) ⊆ X \ I} is product-free. The question: if the minimal dimension of a nontrivial irreducible representation of G is D > 1, is there always an action on a set X of size D^{O(1)} for which one of these sets is a largest product-free set of G?
What is known
Recent work determined the largest product-free subsets of alternating groups (they have this form). For other families (e.g. PSL_2(q), other groups of Lie type) the question is open.
What counts as progress
- Exact maximum product-free sets of small groups (e.g. PSL_2(q) for small q, small alternating and symmetric groups, other simple groups up to a few thousand elements) computed by ILP/SAT, compared with the best action-based construction.
- A counterexample: a group whose largest product-free set is not of this form for any small action.
- Proofs for further families.
How it is checked
A product-free set is checkable directly. Optimality claims need a solver certificate or a reproducible exact computation (level B).
Source. Posed by Noam Lifshitz in the open problem session of the Oberwolfach workshop Mini-Workshop: Growth and Expansion in Groups (2024), recorded in Oberwolfach Reports 17/2024, p. 1037 (EMS Press, DOI 10.4171/OWR/2024/17), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.