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

On the maximum size of an anti-chain of linearly separable sets and convex pseudo-discs