Using the BorsukUlam Theorem
Lectures on Topological Methods in Combinatorics and Geometry
Authors: Matousek, Jiri
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"The "Kneser conjecture"  posed by Martin Kneser in 1955 in the Jahresbericht der DMV  is an innocentlooking problem about partitioning the ksubsets of an nset into intersecting subfamilies. Its striking solution by L. Lovász featured an unexpected use of the BorsukUlam theorem, that is, of a genuinely topological result about continuous antipodal maps of spheres.
Matousek's lively little textbook now shows that Lovász' insight as well as beautiful work of many others (such as Vrecica and Zivaljevic, 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 Matousek 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! […]"
Günter M.Ziegler (Berlin)
Zbl. MATH Volume 1060 Productionsno.: 05001  Reviews

From the reviews:
[...]Matousek's lively little textbook now shows that Lovász' insight as well as beautiful work of many others (such as Vreci´ca and Zivaljevi´c, and Sarkaria) have opened up an exciting area of matheamtics 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 Matousek 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! [..]
Günter M. Ziegler (Berlin)
Zbl. MATH Volume 1060 Productionsno.: 05001
...."The book contains more than 100 exercises, many of them being compressed outlines of interesting results. As such, most of them are both challenging and interesting.
I think this book will be of interest mostly for mathematicians and graduate students. It can certainly be used in a graduate course or seminar. It has an extensive list of references and it covers many interesting and notsoeasy results, with proofs that are sometimes easier than in the original papers. I wouldn't say that this book is an "easy read", but it is very well written, very interesting, and very informative".
Mihaela Poplicher, MAA Online
"This book is intended to make elementary topological methods more accessible to those who work in other areas of mathematics. … This excellent book is useful for specialists in discrete geometry, combinatorics and computer science who want to learn how algebraic topology can be used in their discipline. It is also very suitable as a textbook for such a course." (Ferenc Fodor, Acta Scientiarum Mathematicarum, Vol. 71, 2005)
"The book is addressed to mathematicians, scientists, and engineers working on signal and image processing and medical imaging. It is written by experts and for experts, but each chapter has an introductory part written for nonspecialists, giving them the possibility to find what the chapter is dealing with. The book contains important contributions to the areas mentioned in the title: sampling, wavelets and tomography." (S. Cobzas, Studia universitatis BabesBolyai Mathematica, Vol. XLIX (2), 2004)
"This book … is based on a couple of graduate courses in ‘topological combinatorics’ taught by the author … . The book contains more than 100 exercises, many of them being compressed outlines of interesting results. As such, most of them are both challenging and interesting. … It has an extensive list of references and it covers many interesting and nosoeasy results, with proofs that are sometimes easier than in the original papers. … it is very well written, very interesting, and very informative." (Mihaela Poplicher, MAA Online, January, 2005)
"This excellent textbook is based on lecture notes by the author on applications of some topological methods in combinatorics and geometry. … All chapters include exercises and historical and bibliographical notes. Many hints for further reading are given. … topologists will find here some nice applications for their lectures." (Zdzislaw Dzedzej, Mathematical Reviews, 2004 i)
"Matoušek’s lively little textbook now shows that Lovász’ insight as well as beautiful work of many others … have opened up an exciting area of mathematics that connects combinatorics, graph theory, algebraic topology and discrete geometry. … 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!" (Günter M. Ziegler, Zentralblatt MATH, Vol. 1016, 2003)
"This textbook explains elementary but powerful topological methods based on the BorsukUlam theorem and its generalizations. It covers many substantial results, sometimes with proofs simpler than those in the original papers." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 49 (12), 2003)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Using the BorsukUlam Theorem
 Book Subtitle
 Lectures on Topological Methods in Combinatorics and Geometry
 Authors

 Jiri Matousek
 Series Title
 Universitext
 Copyright
 2003
 Publisher
 SpringerVerlag Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 eBook ISBN
 9783540766490
 DOI
 10.1007/9783540766490
 Softcover ISBN
 9783540003625
 Series ISSN
 01725939
 Edition Number
 1
 Number of Pages
 XII, 214
 Topics