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

TL;DR

This paper investigates the maximum size of anti-chains formed by intersections of points with half-planes and convex pseudo-discs, establishing near-quadratic bounds and equivalences with line arrangements.

Contribution

It provides the first asymptotic bounds for anti-chains of convex pseudo-discs and links these bounds to arrangements of lines and monotone paths.

Findings

Maximum anti-chain size is close to quadratic in n.

Established the equivalence with maximum monotone paths in line arrangements.

Precisely determined the asymptotic bound for anti-chains of 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.