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

Abstract

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.Comment: 10 pages, 3 figures. revised version correctly attributes the idea of Section 3 to Tverberg; and replaced k-sets by "linearly separable sets" in the paper and the title. Accepted for publication in Israel Journal of Mathematic