The Alspach-Liversidge conjecture asserts that every subset of the nonzero elements of a finite Abelian
group can be ordered so that its nonempty partial
sums are pairwise distinct (Alspach and Liversidge 2020). For an ordering 
,
these sums are 
for 
. The empty partial
sum is not included, so a final sum of zero is permitted.
Nguyen et al. (2026, Theorem 1.5) prove that there is an absolute constant 
such that the assertion holds whenever the subset has
at least 
elements. The threshold is independent of the ambient Abelian
group. The theorem does not settle all remaining small cardinalities. Nguyen
et al. (2026) credit ChatGPT with assistance in adapting the absorption argument
and take responsibility for the mathematics.
