A Journey Through Discrete Mathematics pp 221-271 | Cite as

# Using Brouwer’s Fixed Point Theorem

## Abstract

Brouwer’s fixed point theorem from 1911 is a basic result in topology—with a wealth of combinatorial and geometric consequences. In these lecture notes we present some of them, related to the game of HEX and to the piercing of multiple intervals. We also sketch stronger theorems, due to Oliver and others, and explain their applications to the fascinating (and still not fully solved) evasiveness problem.

## Notes

### Acknowledgements

We are grateful to Marie-Sophie Litz and to the referees for very careful reading and a great number of very valuable comments and suggestions on the manuscript. Thanks to Moritz Firsching and Stephen D. Smith, and in particular to Penny Haxell, for additional references and very helpful explanations.

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