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arXiv · 0707.0311

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

Rom Pinchasi·Günter Rote
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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.

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Rom Pinchasi, Günter Rote. 2008-01-08. On the maximum size of an anti-chain of linearly separable sets and convex pseudo-discs. https://doi.org/10.1007/s11856-009-0076-z

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