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Progression-Free Set in a Group

A progression-free set in a group is a subset containing no nontrivial sequence (x,xy,xy^2), where x,y belong to the group and nontrivial means y!=e, with e the identity element (Solymosi 2013, Sabatini 2026). In an additively written Abelian group, this is the arithmetic progression (x,x+y,x+2y). The first and last terms may coincide when y has order 2. Thus a group in which every element other than the identity element has order 2 admits no such progression-free subset of more than one element.

For a finite group G, let r_3(G) be the largest cardinality of a progression-free subset. Sabatini (2026) proved the asymptotic upper bound

r_3(G)<=C|G|exp((-c(ln|G|)^(1/6))/((lnln|G|)^(1/3)))

as |G|->infty, for absolute constants c,C>0. The result applies to arbitrary finite groups, including noncommutative ones.

See also

Arithmetic Progression, Finite Group, Nonaveraging Sequence

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References

Sabatini, L. "Progression-Free Sets in Class 2 and Arbitrary Finite Groups." 10 Nov 2026. https://arxiv.org/abs/2611.14464v1.Solymosi, J. "Roth-Type Theorems in Finite Groups." Europ. J. Combin. 34, 1454-1458, 2013. https://doi.org/10.1016/j.ejc.2013.05.027.

Cite this as:

Weisstein, Eric W. "Progression-Free Set in a Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Progression-FreeSetinaGroup.html

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