Издательство Cambridge University Press, 1986, -187 pp.
This book is an expanded account of a first-year graduate course in combinatorics, given at Louisiana State University, Baton Rouge during the fall semester of 1985.
The traditional ingredients of an initial combinatorics course seem to be combinatorial identities and generating functions, with some introductory design theory or graph theory. Usually, set systems, hyper- graphs and sets of vectors are barely mentioned. But these topics are as worthy of consideration as any, in view of their fundamental nature and elementary structure.
The properties of the subsets of a finite set must lie at the heart of any study of combinatorial theory. This was the motivation for the course given at LSU. It is hoped that combinatorics courses of this nature will be given from time to time, alteating with others of a more standard kind. The book contains considerably more material than one could reasonably hope to cover in a one semester course: this gives the lecturer ample freedom to slant the lectures to his taste.
The book is aimed at future combinatorialists and other mathematicians alike — in addition to the combinatorialists, those specializing in analysis and probability theory are most likely to find the material stimulating.
The contents of the book are made fairly clear by the list of section headings. We look at the very basic features of subsets, such as containment, disjointness, or specified intersection and partition properties. Many of the results have immediate and beautiful applications in combinatorial probability and in analysis. In addition to a thorough grounding in the theory of set systems, the reader will be introduced to some of the basic results in matroid theory, the theory of designs and the Ramsey theory of infinite sets. The reader can consolidate his understanding of the material by tackling over one hundred classroom-tested exercises, ranging from the simple to the challenging. This treatment is by no means comprehensive; it is inevitable that the selected material reflects the taste of the author. Emphasis has been given to theorems with elegant and beautiful proofs; those which may be called the gems of the theory.
Many of the theorems have considerable extensions often of a rather technical nature. In order to preserve the leisurely pace and chatty style, these extensions are not presented. Accordingly, the book's hundred or so references are also far from being comprehensive. Very few special terms are employed in this subject and all the necessary ones are defined. However, some concepts from graph theory are mentioned in passing, occasionally without explanation, to avoid introducing unnecessarily dull passages into the narrative. These concepts are for illustration, though, and are not essential. In any case it is expected that most readers of the book and most students attending a course on it will have encountered elementary graph theory beforehand.
Preface.
Notation.
Representing Sets.
Speer Systems.
The Littlewood-Offord Problem.
Shadows.
Random Sets.
ntersecting Hypergraphs.
The Turan Problem.
Saturated Hypergraphs.
Well-Separated Systems.
Helly Families.
Hypergraphs with a given number of Disjoint Edges.
ntersecting Families.
Factorizing Complete Hypergraphs.
Weakly Saturated Hypergraphs.
soperimetric Problems.
The Trace of a Set System.
tioning Sets of Vectors.
The Four Functions Theorem.
nfinite Ramsey Theory.
This book is an expanded account of a first-year graduate course in combinatorics, given at Louisiana State University, Baton Rouge during the fall semester of 1985.
The traditional ingredients of an initial combinatorics course seem to be combinatorial identities and generating functions, with some introductory design theory or graph theory. Usually, set systems, hyper- graphs and sets of vectors are barely mentioned. But these topics are as worthy of consideration as any, in view of their fundamental nature and elementary structure.
The properties of the subsets of a finite set must lie at the heart of any study of combinatorial theory. This was the motivation for the course given at LSU. It is hoped that combinatorics courses of this nature will be given from time to time, alteating with others of a more standard kind. The book contains considerably more material than one could reasonably hope to cover in a one semester course: this gives the lecturer ample freedom to slant the lectures to his taste.
The book is aimed at future combinatorialists and other mathematicians alike — in addition to the combinatorialists, those specializing in analysis and probability theory are most likely to find the material stimulating.
The contents of the book are made fairly clear by the list of section headings. We look at the very basic features of subsets, such as containment, disjointness, or specified intersection and partition properties. Many of the results have immediate and beautiful applications in combinatorial probability and in analysis. In addition to a thorough grounding in the theory of set systems, the reader will be introduced to some of the basic results in matroid theory, the theory of designs and the Ramsey theory of infinite sets. The reader can consolidate his understanding of the material by tackling over one hundred classroom-tested exercises, ranging from the simple to the challenging. This treatment is by no means comprehensive; it is inevitable that the selected material reflects the taste of the author. Emphasis has been given to theorems with elegant and beautiful proofs; those which may be called the gems of the theory.
Many of the theorems have considerable extensions often of a rather technical nature. In order to preserve the leisurely pace and chatty style, these extensions are not presented. Accordingly, the book's hundred or so references are also far from being comprehensive. Very few special terms are employed in this subject and all the necessary ones are defined. However, some concepts from graph theory are mentioned in passing, occasionally without explanation, to avoid introducing unnecessarily dull passages into the narrative. These concepts are for illustration, though, and are not essential. In any case it is expected that most readers of the book and most students attending a course on it will have encountered elementary graph theory beforehand.
Preface.
Notation.
Representing Sets.
Speer Systems.
The Littlewood-Offord Problem.
Shadows.
Random Sets.
ntersecting Hypergraphs.
The Turan Problem.
Saturated Hypergraphs.
Well-Separated Systems.
Helly Families.
Hypergraphs with a given number of Disjoint Edges.
ntersecting Families.
Factorizing Complete Hypergraphs.
Weakly Saturated Hypergraphs.
soperimetric Problems.
The Trace of a Set System.
tioning Sets of Vectors.
The Four Functions Theorem.
nfinite Ramsey Theory.