Almost-spanning rearrangeable subsets in arbitrary groups

Construct, for every finite group \(\Gamma\) of order n and every subset \(S\subseteq\Gamma\) of size d, a rearrangeable subset \(S'\subseteq S\) of size \(d-o(d)\).

Background

The paper establishes that arbitrary subsets of arbitrary groups admit orderings with almost all partial products distinct, and it obtains rearrangeable subsets of size (1−o(1))d(1-o(1))d in several special regimes.

The stronger formulation asks for an actually rearrangeable subset containing all but a vanishing proportion of the original set. The authors explicitly note that this is already open for Zp\mathbb{Z}_p.

References

Proving a similar result without a density assumption would be of great interest. 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 6, Section 6, “Rearrangeable subsets in groups”

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 \ref{prob:mainconc}, Section 6, Concluding remarks