Introduction to Combinatorial Methods in Geometry

✍ Scribed by Alexander Kharazishvili

• Publisher: Chapman and Hall/CRC
• Year: 2024
• Tongue: English
• Leaves: 396
• Edition: 1
• Category: Library

✦ Synopsis

This book offers an introduction to some combinatorial (also, set-theoretical) approaches and methods in geometry of the Euclidean space Rm. The topics discussed in the manuscript are due to the field of combinatorial and convex geometry.

The author’s primary intention is to discuss those themes of Euclidean geometry which might be of interest to a sufficiently wide audience of potential readers. Accordingly, the material is explained in a simple and elementary form completely accessible to the college and university students. At the same time, the author reveals profound interactions between various facts and statements from different areas of mathematics: the theory of convex sets, finite and infinite combinatorics, graph theory, measure theory, classical number theory, etc.

All chapters (and also the five Appendices) end with a number of exercises. These provide the reader with some additional information about topics considered in the main text of this book. Naturally, the exercises vary in their difficulty. Among them there are almost trivial, standard, nontrivial, rather difficult, and difficult. As a rule, more difficult exercises are marked by asterisks and are provided with necessary hints.

The material presented is based on the lecture course given by the author. The choice of material serves to demonstrate the unity of mathematics and variety of unexpected interrelations between distinct mathematical branches.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
CHAPTER 1: The index of an isometric embedding
CHAPTER 2: Maximal ot-subsets of the Euclidean plane
CHAPTER 3: The cardinalities of at-sets in a real Hilbert space
CHAPTER 4: Isosceles triangles and it-sets in Euclidean space
CHAPTER 5: Some geometric consequences of Ramsey’s combinatorial theorem
CHAPTER 6: Convexly independent subsets of infinite sets of points
CHAPTER 7: Homogeneous coverings of the Euclidean plane
CHAPTER 8: Three-colorings of the Euclidean plane and associated triangles of a prescribed type
CHAPTER 9: Chromatic numbers of graphs associated with point sets in Euclidean space
CHAPTER 10: The Szemerédi–Trotter theorem and its applications
CHAPTER 11: Minkowski’s theorem, number theory, and nonmeasurable sets
CHAPTER 12: Tarski’s plank problem
CHAPTER 13: Borsuk’s conjecture
CHAPTER 14: Piecewise affine approximations of continuous functions of several variables and Caratheodory–Gale polyhedra
CHAPTER 15: Dissecting a square into triangles of equal areas
CHAPTER 16: Geometric realizations of finite and infinite families of sets
CHAPTER 17: A geometric form of the Axiom of Choice
APPENDIX 1: Convex sets in real vector spaces
APPENDIX 2: Real-valued convex functions
APPENDIX 3: The Principle of Inclusion and Exclusion
APPENDIX 4: The Erdös–Mordell inequality
APPENDIX 5: Some facts from graph theory
BIBLIOGRAPHY
INDEX