Piercing and Covering Results in Combinatorial Geometry

Abstract

Combinatorial geometry is a branch of mathematics that studies the arrangement, properties and relationships of geometric objects based on their combinatorial structures. In this thesis, we have mainly studied the following two types of problems in combinatorial geometry: (I) Geometric Transversal Theory and (II) Covering Subsets of the Hypercube with Nice Geometric Objects. (I) Geometric Transversal Theory: Suppose F is a collection of subsets of Rd and T is a family of geometric objects in Rd. For example, T can be a set of points or lines or hyperplanes etc. Then T is said to be a transversal of F if for all F ∈ F there exists T ∈ T such that F ∩T ̸= /0. In other words, we say T pierces or stabs F. For any n ∈ N, F is said to be n-pierceable, if F has a transversal T of size at most n. Helly’s theorem is a cornerstone result in geometric transversal theory. The theorem says that, if we are given a family F of compact convex sets in Rd such that every d+1 sets of F is pierceable by a point then the whole family is pierceable by a single point. Over the century numerous studies have been done by changing the framework of Helly’s theorem from different aspects. Some of the most important variants of Helly’s theorem are Fractional Helly theorem, Colorful Helly theorem, (p,q)-theorem etc. Holmsen and Lee (Israel Journal of Mathematics, 2021) showed that in Rd, colorful Helly theorem implies fractional Helly theorem. Besides these, studying Helly-type theorems for piercing with higher dimensional transversals (for example, lines, hyperplanes or k-dimensional affine spaces, namely k-flats) or n-pierceabily (n > 1) or some special class of sets (for example, axis-parallel boxes, unit disks etc.) is also a common practice. Danzer and Grünbaum (Combinatorica, 1982) gave the first Helly-type result for multiperceability of boxes. In Chapter 3 we have studied a colorful version of their result. One of the most interesting features of our findings is that there is a strict separation between the monochromatic Helly-type result by Danzer and Grünbaum and our colorful Helly-type result. Keller and Perles first extended the (p,q)-theorem to infinite settings, namely (ℵ0,k+2)-theorem for k-transversals (that is, piercing with k-flats). In Chapter 4 we have studied a colorful (ℵ0,2)-theorem for axis parallel boxes piercing with axis parallel lines and k-flats. One thing to notice here is that all these Helly-type results are extremely dependent on the dimension of the ambient space. Adiprasito et al. (Discrete & Computational Geometry, 2020) proved the first dimension independent Helly theorem. In Chapter 5 we have proved a dimension independent colorful Helly theorem for higher dimensional transversals. (II) Covering Subsets of the Hypercube with Nice Geometric Objects: There is a long line of research, spanning over three decades, on problems about covering the vertices of the n-dimensional hypercube Qn = {0,1}n by hyperplanes. Suppose we want to cover all the vertices of Qn with the minimum number of (affine) hyperplanes (a hyperplane H covers a vertex v of Qn if v lies on H). Then at least 2 hyperplanes are required and sufficient also (for example, xi = 0 and xi = 1, for any i ∈ [n]). But what if we want to cover all but one, say the origin, vertices of Qn keeping the origin as uncovered? The celebrated result of Alon and Füredi shows that at least n hyperplanes will be required. Also, observe that the hyperplanes, xi = 1, for all i ∈ [n] are sufficient. Lying in the intersection of finite geometry and extremal combinatorics, numerous variants of this covering problem have been studied. Notice that we can also ask the same question with a slight modification. What will be the minimum degree of a polynomial P ∈ R[x1, . . . ,xn] such that P vanishes at all the vertices of Qn except at the origin, and at the origin P does not vanish at all? As hyperplanes are nothing but multi-linear polynomials, clearly, the size of the hyperplane cover (that is, the minimum number of hyperplanes required for the covering) serves as an upper bound for the size of polynomial cover (that is, the minimum degree of the polynomial that does the covering). Alon and Füredi showed that any polynomial that vanishes at every vertex of the hypercube Qn except the origin and does not vanish at the origin has degree at least n. And hence we cannot cover the vertices with less than n hyperplanes. Since then it has been a question of interest for which forbidden set there is a separation between polynomial covering and hyperplane covering. We have shown that there is strict separation between polynomial covering and hyperplane covering when we consider covering with multiplicities. (We say P ∈ R[x1, . . . ,xn] covers a vertex v of Qn with multiplicity t if v is a zero of P with multiplicity t.)