Geometric Etudes in Combinatorial Mathematics

by Alexander Soifer

-Forewords to the Second Edition.-Forewords to the First Edition.-Preface to the Second Edition.-Preface to the First Edition.-Part I. Original Etudes.-1. Tiling a Checker Rectangle.- 1.1 Introduction.-1.2 Tiling Rectangles by Trominoes.- 1.3 Tetrominoes and 'Color' Reasoning.-1.4Tiling by Linear Polyominoes.- 1.5 Polyominoes and Rotational.-1.6 Symmetries.-1.7 Tiling on Other Surfaces.- 2. Proofs of Existence.- 2.1 The Pigeonhole Principle in Geometry.- 2.2 An Infinite Flock of Pigeons.-3. A Word About Graphs.-3.1 Combinatorics of Acquaintance, or Introduction to Graph Theory. -3.2 More About Graphs.-3.3 Planarity.-3.4 The Intersection Index and the Jordan Curve Theorem.-4. Ideas of Combinatorial Geometry.-4.1 What are Convex Figures?.-4.2 Decomposition of Figures Into Parts of Smaller Diameters.-4.3 Figures of Constant Width.-4.4 Solution of the Borsuk Problem for Figures in the Plane.-4.5 Illumination of Convex Figures.-4.6 Theorems of Helly and Szökefalvi-Nagy.-Part II. New Landscape or the View 18 Years Later.-5. Mitya Karabash and a Tiling Conjecture.-6. Norton Starr's 3-Dimensional Tromino Tiling.-7. Large Progress in Small Ramsey Numbers.-8. The Borsuk Problem Conquered.-9. Etude on the Chromatic Number of the Plane.-10. Farewell to the Reader.-References.-Notation.-Index.

236 pages, Paperback

First published April 1, 1991

About the author

I was once born in Moscow.
I was born free in a country that did not tolerate free men.
And so I left as a refugee, and earned my freedom in the US of A.
Freedom is easy to obtain: just allow it to yourself. However, in order to keep it, one must exercise it, daily.