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A Tribute to Paul Erdos
  • Edited by A. Baker, B. Bollobás, A. Hajnal
  • Published online:
    05 March 2012
    Print publication:
    13 December 1990
  • This volume is dedicated to Paul Erdos, who profoundly influenced mathematics in the twentieth century, with over 1200 papers in number theory, complex analysis, probability theory, geometry, interpretation theory, algebra set theory and combinatorics. One of Erdos' hallmarks was the host of stimulating problems and conjectures, to many of which he attached monetary prices, in accordance with their notoriety. A feature of this volume is a collection of some 50 outstanding unsolved problems, together with their 'value'! Eminent mathematicians from around the world have contributed articles to this volume that reflect the diversity of Erdos' interests, and it will be a fund of insight for number theorists, combinatorialists, set theorists and analysts.

Surveys in Combinatorics
  • Edited by B. Bollobás
  • Published online:
    17 March 2010
    Print publication:
    09 August 1979
  • Combinatorics is an active field of mathematical study and the British Combinatorial Conference, held biennially, aims to survey the most important developments by inviting distinguished mathematicians to lecture at the meeting. The contributions of the principal lecturers at the Seventh Conference, held in Cambridge, are published here and the topics reflect the breadth of the subject. Each author has written a broadly conceived survey, not limited to his own work, but intended for wide readership. Important aspects of the subject are emphasized so that non-specialists will find them understandable. Topics covered include graph theory, matroids, combinatorial set theory, projective geometry and combinatorial group theory. All those researching into any aspect of Combinatorics and its applications will find much in these articles of use and interest.

Results and Problems in Combinatorial Geometry
  • Vladimir G. Boltjansky, Israel Gohberg
  • Translated by B. Bollobás, A. Harris
  • Published online:
    23 October 2009
    Print publication:
    10 October 1985
  • In this short book, the authors discuss three types of problems from combinatorial geometry: Borsuk's partition problem, covering convex bodies by smaller homothetic bodies, and the illumination problem. They show how closely related these problems are to each other. The presentation is elementary, with no more than high-school mathematics and an interest in geometry required to follow the arguments. Most of the discussion is restricted to two- and three-dimensional Euclidean space, though sometimes more general results and problems are given. Thus even the mathematically unsophisticated reader can grasp some of the results of a branch of twentieth-century mathematics that has applications in such disciplines as mathematical programming, operations research and theoretical computer science. At the end of the book the authors have collected together a set of unsolved and partially solved problems that a sixth-form student should be able to understand and even attempt to solve.

  • 8 - Metrics for sparse graphs

  • Edited by Sophie Huczynska, University of St Andrews, Scotland, James D. Mitchell, University of St Andrews, Scotland, Colva M. Roney-Dougal, University of St Andrews, Scotland
  • Book:
    Surveys in Combinatorics 2009
    Published online:
    05 May 2013
    Print publication:
    09 July 2009, pp 211-288

  • Summary

    Abstract

    Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e. each symmetric measurable function from [0, 1]2 to the non-negative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for large-scale real-world networks.

    A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with n vertices and Θ(n2) edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of graphons, which are essentially bounded kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasi-random graphs are in a sense completely general: any sequence of dense graphs contains such a subsequence. Alternatively, their results show that certain natural models of dense inhomogeneous random graphs (one for each graphon) cover the space of dense graphs: there is one model for each point of the completion, producing graphs that converge to this point.

  • ESSENTIALLY INFINITE COLOURINGS OF GRAPHS

  • B. BOLLOBÁS, Y. KOHAYAKAWA, R. H. SCHELP
  • Journal:
    Journal of the London Mathematical Society / Volume 61 / Issue 3 / June 2000
    Published online by Cambridge University Press:
    01 June 2000, pp. 658-670
    Print publication:
    June 2000

  • The classical canonical Ramsey theorem of Erdős and Rado states that, for any integer q [ges ] 1, any edge colouring of a large enough complete graph contains one of three canonically coloured complete subgraphs of order q. Of these canonical subgraphs, one is coloured monochromatically while each of the other two has its edge set coloured with many different colours. The paper presents a condition on colourings that, roughly speaking, requires them to make effective use of many colours (‘essential infiniteness’); this condition is then shown to imply that the colourings in question must contain large refinements of one of two ‘unavoidable’ colourings that are rich in colours. As it turns out, one of these unavoidable colourings is one of the canonical colourings of Erdős and Rado, and the other is a ‘bipartite variant’ of this colouring.

  • Foreword

  • Vladimir G. Boltjansky, Israel Gohberg
  • Translated by B. Bollobás, A. Harris
  • Book:
    Results and Problems in Combinatorial Geometry
    Published online:
    23 October 2009
    Print publication:
    10 October 1985, pp vii-viii

  • Summary

    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.

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