Some new results about a conjecture by Brian Alspach

Simone Costa, Marco Antonio Pellegrini

Published 2020-03-12Version 1

In this short paper we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset $A$ of $\mathbb{Z}_n\setminus \{0\}$ of size $k$ such that $\sum_{z\in A} z\not= 0$, it is possible to find an ordering $(a_1,\ldots,a_k)$ of the elements of $A$ such that the partial sums $s_i=\sum_{j=1}^i a_j$ are nonzero and $s_i\neq s_j$ for all $1 \leq i < j \leq k$. This conjecture is known to be true for subsets of size $k\leq 11$ in cyclic groups of prime order. We extend such result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in $\mathbb{Z}_n$. Finally, we illustrate some connections with other related conjectures.