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Matching binary convexities. (English) Zbl 0543.52001

A collection \({\mathcal C}\) of subsets of a set X is called a convexity if (i) all singletons and \(\emptyset \in {\mathcal C}\), (ii) \({\mathcal C}\) is closed under intersection, (iii) the union of every upward-directed subfamily is also in \({\mathcal C}\). A convexity structure \((X,{\mathcal C})\) is called binary if every (pairwise) intersecting subfamily of \({\mathcal C}\) has a non-empty intersection. The Radon number r(X,\({\mathcal C})\) is defined as the maximal integer r for which there exists an \(A\subset X\), \(| A| =r\) such that the convex hulls of every two disjoint subsets of A are disjoint. (Note that this differs from the usual definition by 1.) A topological convexity structure is said to be normal provided that for every two disjoint closed convex sets A,\(B\in {\mathcal C}\) one can choose closed convex sets \(A',B'\in {\mathcal C}\) satisfying \(A\cup B=X\), \(A\cap B'=A'\cap B=\emptyset.\) In an earlier paper by the same author [Topology Appl. 16, 207-235 (1983; Zbl 0543.52001)] it was proved that the Radon number of an n-dimensional continuum with compact intervals and with a normal binary convexity is equal to \(r_ n\) or \(r_ n+1\), where \(r_ n\) denotes the Radon number of the n-cube with the subcube convexity. The paper under review determines exactly those dimensions \(n\) in which there exist normal binary continuum having Radon number \(r_ n+1\). For the constructions a matching procedure is developed.
Reviewer: J.Pach

MSC:

52A01 Axiomatic and generalized convexity
52A35 Helly-type theorems and geometric transversal theory
05C10 Planar graphs; geometric and topological aspects of graph theory

Citations:

Zbl 0543.52001
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