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

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.

💡 Research Summary

The paper investigates the extremal size of antichains formed by subsets of a fixed planar point set P (|P| = n) that are obtained as intersections with half‑planes, and extends the study to families of convex pseudo‑discs. An antichain is a collection of sets none of which contains another. Classical order‑theoretic results (Sperner, Dilworth) give general bounds for antichains, but they do not exploit the geometric constraints imposed by linear separability. The authors bridge this gap by showing that the problem of maximizing the size of such an antichain is equivalent to the problem of finding the longest monotone path in an arrangement of n lines.

Main contributions

1. Half‑plane antichains – By interpreting each half‑plane as a sweeping line that partitions the point set, the authors construct a directed acyclic graph (DAG) whose vertices correspond to line‑intersection events in the arrangement. Two subsets are comparable exactly when there is a directed path between their corresponding vertices. Consequently, the size of a maximal antichain equals the length of a longest antichain‑free (i.e., mutually incomparable) chain in this DAG, which is precisely the length of a longest monotone path. For a generic arrangement the longest monotone path has length Θ(n²); a concrete construction using two families of parallel lines yields a lower bound of (1/4)n² and an obvious upper bound of n². Hence the maximum antichain size among half‑plane intersections is asymptotically quadratic.

2. Convex pseudo‑disc antichains – A convex pseudo‑disc is defined as a set A = P ∩ C where C is a convex region, with the additional requirement that for any two such sets A and B the set difference conv(A) \ conv(B) ∪ conv(B) \ conv(A) is non‑empty and connected. This condition guarantees that A and B overlap partially but neither contains the other, mirroring the antichain property in a geometric fashion. Using the same line‑arrangement reduction, the authors prove an upper bound of (1/4)n² + O(n) for the size of a maximal pseudo‑disc antichain. They complement this with an explicit construction: place the n points uniformly on a circle and select half‑planes defined by tangents to the circle. Each tangent yields a distinct subset, and any two such subsets satisfy the pseudo‑disc condition while remaining incomparable. The construction produces exactly ⌊n/2⌋·⌊n/2⌋ ≈ n²/4 sets, establishing the lower bound and confirming the asymptotic tightness.

Technical insights

• The reduction to monotone paths reveals a deep connection between order‑theoretic antichains and geometric sweep‑line processes.
• The connected‑difference condition for pseudo‑discs is a subtle geometric strengthening that still permits a quadratic number of mutually incomparable sets.
• The paper provides both combinatorial arguments (using Dilworth‑type reasoning on the DAG) and constructive geometric examples, thereby covering both existence and optimality.

Implications and future work
The results have several ramifications. In machine learning, the size of a half‑plane antichain reflects the richness of linearly separable labelings that a single linear classifier can realize on a fixed dataset; a quadratic bound indicates a high expressive capacity. In computational geometry, the findings suggest that arrangements of lines can encode large families of independent convex regions, which may be useful for range‑search structures or for designing worst‑case inputs for algorithms that rely on hierarchical decompositions. The authors propose extensions to higher dimensions (hyperplane arrangements leading to Θ(n^d) bounds), to non‑linear separators (circles, ellipses, polynomial curves), and to dynamic settings where points are inserted or deleted while maintaining a maximal antichain.

In summary, the paper establishes that the maximal antichain of linearly separable subsets of an n‑point planar set is Θ(n²), and that for the more restrictive family of convex pseudo‑discs the exact asymptotic constant is 1/4. The work elegantly unifies order‑theoretic concepts with geometric sweep‑line techniques, providing tight bounds and opening several avenues for further exploration in combinatorial geometry and related algorithmic fields.