Graham's rearrangement conjecture in finite abelian groups

Abstract

Graham conjectured in 1971 that every subset of the nonzero elements of a prime cyclic group admits an ordering with pairwise distinct nonempty partial sums. Recently, the work of Pham and Sauermann, together with earlier results, settled this conjecture for all sufficiently large primes. In this paper, we prove that there is an absolute integer $n_0$ such that every subset $S\subseteq G\setminus\{0\}$ of a finite abelian group with $|S|\ge n_0$ admits such an ordering, which also solves a conjecture of Alspach and Liversidge for all sufficiently large sets. A key ingredient of our proof is a sufficient condition via additive dimension: If $|S|=n$ and $S$ contains a dissociated subset of size at least $Ω((\log n)^2/\log\log n)$, then $S$ has an ordering in which no proper nonempty interval of consecutive terms sums to zero. This bound is within a factor of $O(\log n/\log\log n)$ of the smallest possible additive dimension of an $n$-element set.