  In this paper, we consider saturation problems related to the celebrated
Erd\H{o}s--Szekeres convex polygon problem. For each $n \ge 7$, we construct a
planar point set of size $(7/8) \cdot 2^{n-2}$ which is saturated for convex
$n$-gons. That is, the set contains no $n$ points in convex position while the
addition of any new point creates such a configuration. This demonstrates that
the saturation number is smaller than the Ramsey number for the
Erd\H{o}s--Szekeres problem. The proof also shows that the original
Erd\H{o}s--Szekeres construction is indeed saturated.
  Our construction is based on a similar improvement for the saturation version
of the cups-versus-caps theorem. Moreover, we consider the generalization of
the cups-versus-caps theorem to monotone paths in ordered hypergraphs. In
contrast to the geometric setting, we show that this abstract saturation number
is always equal to the corresponding Ramsey number.
