  We show that the maximum cardinality of an anti-chain composed of
intersections of a given set of n points in the plane with half-planes is close
to quadratic in n. We approach this problem by establishing the equivalence
with the problem of the maximum monotone path in an arrangement of n lines. For
a related problem on antichains in families of convex pseudo-discs we can
establish the precise asymptotic bound: it is quadratic in n. The sets in such
a family are characterized as intersections of a given set of n points with
convex sets, such that the difference between the convex hulls of any two sets
is nonempty and connected.
