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 $. 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.

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. Advances in Applied Mathematics, 2017, 82, pp.83-101. ⟨10.1016/j.aam.2016.07.004⟩. ⟨hal-01222193v3⟩