Complete Kneser Transversals

TL;DR

This paper investigates the maximum size of point sets in Euclidean space that guarantee the existence of a common transversal plane intersecting convex hulls of all k-subsets, introducing a discrete variant and exploring bounds and exact values.

Contribution

It introduces a discrete version of the Kneser transversal problem, studies its relation to the original, and provides bounds and exact values using oriented matroids and Radon's theorem.

Findings

Established bounds for the discrete Kneser transversal number m*

Derived the asymptotic behavior of m* for cyclic polytopes

Connected m* with chromatic numbers of Kneser hypergraphs

Abstract

Let $k,d,\lambda\geqslant1$ be integers with $d\geqslant\lambda $. Let $m(k,d,\lambda)$ be the maximum positive integer $n$ such that every set of $n$ points (not necessarily in general position) in $\mathbb{R}^{d}$ has the property that the convex hulls of all $k$-sets have a common transversal $(d-\lambda)$-plane. It turns out that $m(k, d,\lambda)$ 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.