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Pré-Publication, Document De Travail Année : 2016

Complete Kneser Transversals

Résumé

Let $k,d,\lambda\geqslant1$ be integers with $d\geqslant\lambda $. In 2010, the following function was introduced: $m(k,d,\lambda)\overset{\mathrm{def}}{=}$ 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 so-called alternating oriented matroid we provide the asymptotic behavior of the function $m^*$ for the family of cyclic polytopes.
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Dates et versions

hal-01222193 , version 1 (29-10-2015)
hal-01222193 , version 2 (26-06-2016)
hal-01222193 , version 3 (10-08-2016)

Identifiants

Citer

Jonathan Chappelon, Leonardo Martínez-Sandoval, Luis Montejano, Luis Pedro Montejano, Jorge Luis Ramírez Alfonsín. Complete Kneser Transversals. 2016. ⟨hal-01222193v2⟩
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