## Abstract

Let k,d,λ⩾1 be integers with d⩾λ. Let m(k,d,λ) be the maximum positive integer n such that every set of n points (not necessarily in general position) in R^{d} has the property that the convex hulls of all k-sets have a common transversal (d−λ)-plane. It turns out that m(k,d,λ) is strongly connected with other interesting problems, for instance, the chromatic number of Kneser hypergraphs and a discrete version of Rado's centerpoint theorem. In the same spirit, we introduce a natural discrete version m^{⁎} of m by considering the existence of complete Kneser transversals. We study the relation between them and give a number of lower and upper bounds of m^{⁎} as well as the exact value in some cases. The main ingredient for the proofs are Radon's partition theorem as well as oriented matroids tools. By studying the alternating oriented matroid we obtain the asymptotic behavior of the function m^{⁎} for the family of cyclic polytopes.

Original language | English |
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Pages (from-to) | 83-101 |

Number of pages | 19 |

Journal | Advances in Applied Mathematics |

Volume | 82 |

DOIs | |

State | Published - 1 Jan 2017 |

Externally published | Yes |

## Keywords

- Cyclic polytope
- Kneser hypergraphs
- Oriented matroids
- Transversals

## ASJC Scopus subject areas

- Applied Mathematics