**University Lecture Series**

Volume: 77;
2021;
148 pp;
Softcover

MSC: Primary 52; 05;

**Print ISBN: 978-1-4704-6709-8
Product Code: ULECT/77**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

**Electronic ISBN: 978-1-4704-6768-5
Product Code: ULECT/77.E**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

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#### Supplemental Materials

# Combinatorial Convexity

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*Imre Bárány*

This book is about the combinatorial properties of convex sets,
families of convex sets in finite dimensional Euclidean spaces, and
finite points sets related to convexity. This area is classic, with
theorems of Helly, Carathéodory, and Radon that go back more than a
hundred years. At the same time, it is a modern and active field of
research with recent results like Tverberg's theorem, the colourful
versions of Helly and Carathéodory, and the \((p, q)\) theorem
of Alon and Kleitman. As the title indicates, the topic is convexity
and geometry, and is close to discrete mathematics. The questions
considered are frequently of a combinatorial nature, and the proofs
use ideas from geometry and are often combined with graph and
hypergraph theory.

The book is intended for students (graduate and undergraduate
alike), but postdocs and research mathematicians will also find it
useful. It can be used as a textbook with short chapters, each
suitable for a one- or two-hour lecture. Not much background is
needed: basic linear algebra and elements of (hyper)graph theory as
well as some mathematical maturity should suffice.

#### Readership

Undergraduate and graduate students and researchers interested in combinatorial properties of convexity and convex sets.

#### Reviews & Endorsements

This is an elegant, well written, concise treatment of an attractive and active subject, written by an expert who has made important contributions to the area himself. I am sure this will be a successful textbook.

-- Noga Alon, Princeton University and Tel Aviv University

I think this book is a gem.

-- János Pach, Rényi Institute of Mathematics, Budapest

#### Table of Contents

# Table of Contents

## Combinatorial Convexity

- Cover Cover11
- Title page iii4
- Copyright iv5
- Contents v6
- Preface vii8
- Basic concepts 110
- Carathéodory’s theorem 918
- Radon’s theorem 1322
- Topological Radon 1726
- Tverberg’s theorem 2130
- General position 2736
- Helly’s theorem 2938
- Applications of Helly’s theorem 3342
- Fractional Helly 3948
- Colourful Carathéodory 4150
- Colourful Carathéodory again 4554
- Colourful Helly 4958
- Tverberg’s theorem again 5362
- Colourful Tverberg theorem 5766
- Sarkaria and Kirchberger generalized 6170
- The Erdős-Szekers theorem 6372
- The same type lemma 6776
- Better bound for the Erdős-Szekeres number 7180
- Covering number, planar case 7786
- The stretched grid 8190
- Covering number, general case 8796
- Upper bound on the covering number 91100
- The point selection theorem 95104
- Homogeneous selection 99108
- Missing few simplices 101110
- Weak 𝜖-nets 105114
- Lower bound on the size of weak 𝜖-nets 109118
- The (𝑝,𝑞) theorem 113122
- The colourful (𝑝,𝑞) theorem 119128
- 𝑑-intervals 123132
- Halving lines, havling planes 127136
- Convex lattice sets 131140
- Fractional Helly for convex lattice sets 137146
- Bibliography 143152
- Index 147156
- Back Cover Back Cover1159