Complete Kneser Transversals

Jonathan Chappelon (1) , Leonardo Martínez-Sandoval (2, 1) , Luis Montejano (2) , Luis Pedro Montejano (1) , Jorge Luis Ramírez Alfonsín (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 Instituto de Matematicas [México]

Let $k,d,\lambda\geqslant1$ be integers with $d\geqslant\lambda $. In 2010, the following function was introduced: \par\smallskip $m(k,d,\lambda)\overset{\mathrm{def}}{=}$ 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. \smallskip\par 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 central Theorem. \par In the same spirit, we introduce a natural discrete version $m^*$ of $m$ by considering the existence of \emph{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 \emph{alternating} oriented matroid we provide the asymptotic behavior of the function $m^*$ for the family of cyclic polytopes.

Mots clés: cyclic polytope.; transversals; Kneser hypergraphs; oriented matroids

Domaines: Combinatoire [math.CO]; Géométrie métrique [math.MG]; Mathématique discrète [cs.DM]

Jonathan Chappelon, Leonardo Martínez-Sandoval, Luis Montejano, Luis Pedro Montejano, Jorge Luis Ramírez Alfonsín. Complete Kneser Transversals. 2015. ⟨hal-01222193v1⟩