There are many elegant results in the theory of convex bodies that may be fully understood by high school students, and at the same time be of interest to expert mathematicians. The aim of this book is to present some of these results. We shall discuss combinatorial problems of the theory of convex bodies, mainly connected with the partition of a set into smaller parts.

The theorems and problems in the book are fairly recent: the oldest of them is just over thirty years old, and many of the theorems are still in their infancy. They were published in professional mathematical journals during the last five years.

We consider the main part of the book to be suitable for high school students interested in mathematics. The material indicated as complicated may be skipped by them. The most straightforward sections concern plane sets: §§1–3, 7–10, 12–14. The remaining sections relate to spatial (and even n-dimensional) sets. For the keen and well-prepared reader, at the end of the book will be found notes, as well as a list of journals, papers and books. References to the notes are given in round brackets (). and references to the bibliography in square brackets []. In several places, especially in the notes, the discussion is at the level of scientific papers. We did not consider it inappropriate to include such material in a non-specialized book.

This book originally appeared in Russian almost twenty years ago; nevertheless it is as fresh now as then. No better exposition of the main results has since appeared, and the problems stated at the end of the book still remain unsolved.

I would like to mention two books which appeared after this volume and which are closely related to this material. The first is “The Decomposition of Figures into Smaller Parts” by the same authors, which appeared in English translation in the University of Chicago Press in 1980, and also the book of V.G. Boltyansky and P.S. Soltan “Combinatorial Geometry of Different Classes of Convex Sets” Stiintsa, Kishinev, 1978 (in Russian). The first book is a popular book devoted only to combinatorial problems of the plane, and the second book is on the level of mathematical research monographs.

Finally, I would like to thank Cambridge University Press and Dr. David Tranah for their interest and cooperation.

The purpose of this paper is to describe a typical communications system and highlight the problems encountered when transmitting signals. We shall discuss how the signal is corrupted during transmission and examine the possibility of a telecommunications system within which the transmitting and receiving terminals continually adapt their strategies to optimise the use of the particular communications channel at their disposal.

There are many ways in which a signal can be corrupted during its transmission from a sending terminal to a receiving terminal. Figure 1 shows a generalised system and indicates where and in what way the signal can be corrupted and, in particular the various forms of noise that can be added to it. Within this generalised setting, the original signal may first be sent to a switching centre. The switching centre will also have a variety of other signals from other terminals arriving simultaneously and a common form of noise/interface is caused by ‘crosstalk’ which involves the ‘leakage’ of one signal to another. The next processing step will probably involve modulation.

There are two main reasons for using modulation. These are firstly to shift the message signal frequencies to a band which can be transmitted efficiently over the channel and secondly to enable the simultaneous transmission of several signals over a single transmission link. In order to understand the significance of this process one must appreciate that if the frequency domain of a time–varying signal is examined one normally finds that most of the information is contained in a finite portion of the frequency spectrum.

INTRODUCTION

There are many situations in which the idea of a “flow” is important, such as

1. – fluid passing through a system of pipes,

2. – electrons moving along the wires of an electrical circuit,

3. – information or rumours spreading through a community,

4. – disease infecting the members of a population.

In all these examples are present the ideas of both space and time. Each requires a medium (generally a set of points with interconnections) and a permeating quantity (such as fluid or disease). In any particular example of one of these four situations, there is a set of rules which govern the consequent flow, and such sets of rules vary a great deal in their styles and degrees of complexity.

This volume contains eight of the nine invited lectures given at the tenth British Combinatorial Conference held in the University of Glasgow, 22 – 26 July, 1985. Although British in name and organisation, these biennial conferences are international in personnel, the nine invited speakers consisting of three from the U.K., three from Europe and three from North America.

Each of the invited speakers was asked to describe developments in a particular branch of combinatorics. The resulting volume provides a broad survey of many areas of contemporary research interest, both theoretical and applied, showing relation between combinatorics and number theory, probability and algebra. I should like to thank the authors for their cooperation in producing their typescripts in accordance with a tight schedule, thus enabling the preparation of this volume to proceed according to plan.

Contributed papers of the conference will be published in a special issue of Ars Combinatoria. This arrangement proved successful at the previous conference at Southampton in 1983.

Finally, thanks are due to the Cambridge University Press for their guidance and cooperation in the preparation of this volume, and to both the British Council and the London Mathematical Society for their financial support of the conference.

Introduction

L. J. Rogers' paper (Rogers; 1894) which contains the Rogers–Ramanujan identities together with their proof was ignored for 20 years before Ramanujan came across it while leafing through old volumes of the Proceedings of the London Mathematical Society. In the interim, Ramanujan had discovered the Rogers–Ramanujan identities empirically, and they were making the rounds as major unsolved problems (cf. Hardy; 1940, p. 91). This is undoubtedly one of the very few times that a set of significant unsolved problems was solved 20 years before it was posed.

The most obvious reason Rogers' paper lay buried is that it is page after page of q–series identities with the Rogers–Ramanujan identities sneaking past in mild disguise on page 10 of this tour de force.

As more discoveries were made, the subject became even less readable. The Rev. F. H. Jackson was one of the early pioneer q–series researchers. His papers also read much like Rogers'. It is not surprising to read in Jackson's obituary (Chaundy (1962)); “Once (with a whimisical smile one imagines) he [Jackson] recounted the occasion of his quarrel with our Society [the L.M.S.]: he had read a paper when someone remarked: ‘Surely, Mr. President, we have heard all this before.’ He strode from the room and never darkened our pages again.” As it turned out this critical remark was directed at what was, in fact, Jackson's most valuable paper.

1. (i) results concerning the structure of the graphs with no minor isomorphic to a fixed graph

2. (ii) results concerning a conjecture of K. Wagner. that for any infinite set of graphs one of its members is isomorphic to a minor of another. and

3. (iii) algorithmic results concerning the DISJOINT CONNECTING PATHS problem.

INTRODUCTION

There are two fundamental questions which motivate the work we report on here.

(A) (K. Wagner's well-quasi-ordering conjecture). Is it true that for every infinite sequence G1, G2, … of graphs, there exist i, j with i < j such that Gi is isomorphic to a minor of Gj?

[Graphs in this paper are finite and may have loops or multiple edges. A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges.]

(B) (The DISJOINT CONNECTING PATHS problem). If k ≥ 0, is there a polynomially-bounded algorithm to decide, given a graph G and vertices s1, …, sk, t1, …, tk of G, whether there are k mutually disjoint paths P1, …, Pk of G where Pi has ends si, ti (1 ≤ i ≤ k)?

[Two paths are disjoint if they have no common vertices.] Some of the background to these questions is discussed in sections 2 and 3.