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Deep Cliques in Point Sets.

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Summary
This summary is machine-generated.

This study introduces k-deep cliques, subsets of points where pairs of points are k-deep. It establishes bounds for k-deep clique sizes in point sets, with specific focus on halving cliques.

Keywords:
CliquesDepth measuresHalving linesTukey depthj-Facetsk-Degenerate graphsk-Sets

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Area of Science:

  • Computational Geometry
  • Combinatorial Geometry
  • Discrete Mathematics

Background:

  • Introduces the concept of k-deep pairs of points within a set P.
  • Defines a k-deep clique as a subset of P where all pairs are k-deep.

Purpose of the Study:

  • To establish tight bounds on the size of k-deep cliques in a set of n points.
  • To investigate the properties and existence of halving cliques (a special case of k-deep cliques).

Main Methods:

  • Analysis of geometric properties of point sets and their subsets.
  • Derivation of lower and upper bounds for k-deep clique sizes.
  • Examination of specific cases, including points in general position and convex position.

Main Results:

  • A lower bound of \(\lceil n/2 \rceil\) for the size of a k-deep clique in a general position point set.
  • An upper bound of \(\lfloor n^2/4 \rfloor\) for the size of a k-deep clique in any point set.
  • Specific bounds for \(k = n/2\) (halving cliques), including necessary conditions on the total number of points for a given halving clique size.

Conclusions:

  • The study provides tight bounds for k-deep clique sizes, demonstrating their existence and limitations.
  • Results for halving cliques offer insights into their structure and relationship with the overall point set.
  • The findings contribute to the understanding of geometric structures within point sets.