On Transversal Problems For Translates

One of the most widely known theorems in Discrete and Combinatorial Geometry is Helly’s Theorem, which states that for a family F of finite convex sets or infinite compact convex sets in Rd (d≥2), if every d+1 members of F have a point in common, then all the members of F have a point in common. Helly’s Theorem spawned numerous generalizations and variants-elly-type problems. These problems usually have the following form:let F be a collection of sets and K be a fixed integer. If every k members of F have property P, then the entire family has property Q.A line transversal to a family of sets in Rd is a straight line having a non-empty intersection with every member of the family. Alternatively, we also say that this family has property T. When P is replaced by property T, this form of Helly-type problem is called line transversal problem. For the sake of convenience, denote the family whose every K-membered subfamily has property T by T(K)-family.In the thesis, we mainly discuss line transversal problems for translates in the plane. In Chapter 1, we introduce the research background and basis concepts of line transversal. In Chapter 2, we prove that every finite T(3)-family F of unit diameter discs, if the distance between the centers of any two members of the family F is greater than 0.95, then there is a parallel strip of width 0.67 intersecting all the members of F. In Chapter 3, applying the conclusion obtained in Chapter 2, we study an open problem posed by Heppes in 2007, and determine an upper bound for Katchalski-Lewis transversal problem about T(3)-families of pairwise disjoint translates of a regular 2n-gon (n≥5,n∈Z+):if 5≤n≤34, then there is a line intersecting all but at most 3 members of F; if n≥35, then there is a line intersecting all but at most 2 members of F. In Chapter 4, we describe the sufficient conditions for the T(3)-families and T(4)-families of disjoint translates of a square having property T, respectively.