Using the Borsuk-Ulam theorem. Lectures on topological methods in combinatorics and geometry. Written in cooperation with Anders Björner and Günter M. Ziegler. (English) Zbl 1016.05001

Universitext. Berlin: Springer. xii, 196 p. (2003).
The “Kneser conjecture” – posed by Martin Kneser in 1955 in the Jahresbericht der DMV – is an innocent-looking problem about partitioning the \(k\)-subsets of an \(n\)-set into intersecting subfamilies. Its striking solution by L. Lovász [J. Comb. Theory, Ser. A 25, 319-328 (1978; Zbl 0418.05028)] featured an unexpected use of the Borsuk-Ulam theorem [K. Borsuk, Fundam. Math. 20, 177-190 (1933; Zbl 0006.42403)], that is, of a genuinely topological result about continuous antipodal maps of spheres.
Matoušek’s lively little textbook now shows that Lovász’ insight as well as beautiful work of many others (such as Vrecića and Živaljević, and Sarkaria) have opened up an exciting area of mathematics that connects combinatorics, graph theory, algebraic topology and discrete geometry. What seemed like an ingenious trick in 1978 now presents itself as an instance of the “test set paradigm”: to construct configuration spaces for combinatorial problems such that coloring, incidence or transversal problems may be translated into the (non-)existence of suitable equivariant maps.
The vivid account of this area and its ramifications by Matoušek is an exciting, a coherent account of this area of topological combinatorics. It features a collection of mathematical gems written with a broad view of the subject and still with loving care for details. Recommended reading!
The reviewer is happy to acknowledge that he has contributed minor input to the writing of this book.


05-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics
55-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology
55M20 Fixed points and coincidences in algebraic topology
52A35 Helly-type theorems and geometric transversal theory
05C15 Coloring of graphs and hypergraphs
05C10 Planar graphs; geometric and topological aspects of graph theory
52-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to convex and discrete geometry
55R80 Discriminantal varieties and configuration spaces in algebraic topology
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)