Abstract
Let \(\mathcal {F}\) be a family of convex sets in \({\mathbb {R}}^d,\) which are colored with \(d+1\) colors. We say that \(\mathcal {F}\) satisfies the Colorful Helly Property if every rainbow selection of \(d+1\) sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family \(\mathcal {F}\) there is a color class \(\mathcal {F}_i\subset \mathcal {F},\) for \(1\le i\le d+1,\) whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension \(d\ge 2\) there exist numbers f(d) and g(d) with the following property: either one can find an additional color class whose sets can be pierced by f(d) points, or all the sets in \(\mathcal {F}\) can be crossed by g(d) lines.
Highlights
1.1 Helly-type theoremsLet F be a finite family of convex sets in Rd
The 1913 theorem of Helly [14] states that a finite family F of convex sets has a non-empty intersection (i.e., F can be pierced by a single point) if and only if each of its subsets F ⊂ F
Alon and Kalai [2] show that the following almost-Helly property holds for k = d − 1: If every d + 1 of the sets of F can be crossed by a hyperplane, F admits a transversal by h hyperplanes, where the number h = h(d) depends only on the dimension d
Summary
Let F be a finite family of convex sets in Rd. We say that a collection X of geometric objects (e.g., points, lines, or k-flats – k-dimensional affine subspaces of Rd) is a transversal to F, or that F can be pierced or crossed by X, if each set of F is intersected by some member of X. The 1913 theorem of Helly [14] states that a finite family F of convex sets has a non-empty intersection (i.e., F can be pierced by a single point) if and only if each of its subsets F ⊂ F of size at most d + 1 can be pierced by a point.
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