Using the Borsuk-Ulam theorem

Important results in combinatorics, discrete geometry, and theoretical computer science have been established using algebraic topology, yet their proofs are often misunderstood and dispersed across research papers. This book serves as the first textbook to systematically present a significant portion of these results, focusing on equivariant methods derived from the Borsuk-Ulam theorem and its generalizations. The text maintains a very elementary approach to topology, avoiding complex concepts like homology theory and homotopy groups. It requires no prior knowledge of algebraic topology, only a foundation in undergraduate mathematics, gradually explaining necessary topological notions. The book delves into substantial combinatorial results, including Kneser's conjecture, from various perspectives. It also surveys the history of the material, references, related results, and advanced methods in separate subsections. Accompanied by numerous exercises of varying difficulty, many of which outline additional results, the book is richly illustrated and includes a detailed index and extensive bibliography. Originating from a graduate course taught in Prague in 1993, the text has evolved through various iterations, including a pre-doctoral course at ETH Zurich in 2001, where most material was covered in approximately 30 hours of teaching.