  In this paper, we address sorting networks that are constructed from
comparators of arity $k > 2$. That is, in our setting the arity of the
comparators -- or, in other words, the number of inputs that can be sorted at
the unit cost -- is a parameter. We study its relationship with two other
parameters -- $n$, the number of inputs, and $d$, the depth.
  This model received considerable attention. Partly, its motivation is to
better understand the structure of sorting networks. In particular, sorting
networks with large arity are related to recursive constructions of ordinary
sorting networks. Additionally, studies of this model have natural
correspondence with a recent line of work on constructing circuits for majority
functions from majority gates of lower fan-in.
  Motivated by these questions, we obtain the first lower bounds on the arity
of constant-depth sorting networks. More precisely, we consider sorting
networks of depth $d$ up to 4, and determine the minimal $k$ for which there is
such a network with comparators of arity $k$. For depths $d=1,2$ we observe
that $k=n$. For $d=3$ we show that $k = \lceil \frac n2 \rceil$. For $d=4$ the
minimal arity becomes sublinear: $k = \Theta(n^{2/3})$. This contrasts with the
case of majority circuits, in which $k = O(n^{2/3})$ is achievable already for
depth $d=3$.
