  For a finite set $A$ of size $n$, an ordering is an injection from
$\{1,2,\ldots,n\}$ to $A$. We present results concerning the asymptotic
properties of the length $L_n$ of the longest arithmetic subsequence in a
random ordering of an additive set $A$. In the torsion-free case where $A =
[1,n]^d\subseteq {\bf Z}^d$, we prove that $L_n\sim 2d\log n/\log\log n$. We
show that the case $A = {\bf Z}/n{\bf Z}$ behaves asymptotically like the
torsion-free case with $d=1$, and then use this fact to compute the expected
length of the longest arithmetic subsequence in a random ordering of an
arbitrary finite abelian group. We also prove that the number of orderings of
${\bf Z}/n{\bf Z}$ without any arithmetic subsequence of length $3$ is
$2^{n-1}$ when $n\geq 2$ is a power of $2$, and zero otherwise. We conclude
with a concrete application to elementary $p$-groups and a discussion of
possible noncommutative generalisations.
