Large rearrangeable subsets in arbitrary groups

Construct, for every finite group Γ of order n and every subset S⊆Γ of size d, a rearrangeable subset S′⊆S of size d−o(d).

Background

The paper’s main general-group result produces an ordering whose partial products contain only a vanishing proportion of repetitions. A stronger approximation would instead find a nearly complete subset that is exactly rearrangeable.

This problem would extend the paper’s dense-regime and special-group results to arbitrary subsets of arbitrary finite groups. The authors explicitly note that it is already unresolved for cyclic groups of prime order.

References

For any group \Gamma of order n and any subset S\subseteq \Gamma of size d, show that there exists a subset S' \subseteq S of size d-o(d) which is rearrangeable. Problem~\ref{prob:mainconc} is already open for cyclic groups \mathbb{Z}_p of prime order.

— Towards Graham's rearrangement conjecture via rainbow paths  (2503.01825 - Bucić et al., 3 Mar 2025) in Problem 1, Section 7 (Concluding remarks), subsection “Rearrangeable subsets in groups”