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Complete Kneser transversals. (English) Zbl 1352.52017

Summary: Let \(k, d, \lambda \geqslant 1\) 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 \(m^\ast\) 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 \(m^\ast\) 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 \(m^\ast\) for the family of cyclic polytopes.

MSC:

52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
52A35 Helly-type theorems and geometric transversal theory
05C15 Coloring of graphs and hypergraphs
52C40 Oriented matroids in discrete geometry
52B99 Polytopes and polyhedra
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References:

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