Helly-type results for k-systems




Pike, Timothy

Journal Title

Journal ISSN

Volume Title



Theorems of the form "if every m objects from a set has a certain property P, then the entire set has the property P" are known as Helly-type theorems, after Eduard Helly's theorem of 1923. Many transversal theorems on collections of bodies in Rn have been stated and proved in this Helly-type style. In this paper, we primarily focus on the planar case of transversals providing support to a collection of convex bodies. Previous authors have shown that given a sufficiently large family of pairwise disjoint convex planar bodies with the property that any three bodies share a support line, the entire family shares at least one support line. Unfortunately, sufficiently large does not provide an exact number of bodies necessary to ensure that if any three bodies have the support property, then the entire family does. Using what is called a (special) set-system of words and letters to model these disjoint convex planar bodies, the existing literature shows us that the minimum number of bodies necessary to ensure the aforementioned support property is no more than 143. In this paper we do not provide the exact number, but rather a generalized framework for which this planar case is a special case. Consistent with earlier authors, we too use a set-system of words and letters. Within our framework, these systems are known as k-systems and special k-systems. We provide a few Helly-type results on both k-systems and special k-systems.



Helly-type, K-system, Convex, Transversal