Graham, Ronald L., Nešetřil, Jaroslav, Butler, Steve (Eds.)
2nd ed. 2013, XIX, 563 p. 44 illus., 2 illus. in color.
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New and extensively revised edition with approximately 100 pages of new material
Contains new articles from prominent mathematicians such as Larry Guth, Alexander Razborov and Joel Spencer
Includes new introductions to all the individual parts, with an update of scientific activity in the last 20 years
The first book bringing together articles surveying the legacy of Paul Erdős in all fields of his activity
Features Paul Erdős' influence in number theory, random structures and geometry, with the most general collection of recent surveys available
This is the most comprehensive survey of the mathematical life of the legendary Paul Erdős (1913-1996), one of the most versatile and prolific mathematicians of our time. For the first time, all the main areas of Erdős' research are covered in a single project. Because of overwhelming response from the mathematical community, the project now occupies over 1000 pages, arranged into two volumes. These volumes contain both high level research articles as well as key articles that survey some of the cornerstones of Erdős' work, each written by a leading world specialist in the field. A special chapter "Early Days", rare photographs, and art related to Erdős complement this striking collection. A unique contribution is the bibliography on Erdős' publications: the most comprehensive ever published. This new edition, dedicated to the 100th anniversary of Paul Erdős' birth, contains updates on many of the articles from the two volumes of the first edition, several new articles from prominent mathematicians, a new introduction, more biographical information about Paul Erdős, and an updated list of publications.
The first volume contains the unique chapter "Early Days", which features personal memories of Paul Erdős by a number of his colleagues. The other three chapters cover number theory, random methods, and geometry. All of these chapters are essentially updated, most notably the geometry chapter that covers the recent solution of the problem on the number of distinct distances in finite planar sets, which was the most popular of Erdős' favorite geometry problems.
Content Level »Research
Keywords »Erdős existence argument - Erdős–Turán - Paul Erdős - Ramsey theory - additive representation functions - extremal theory - incidence problems - sum-product phenomena
VOLUME I.- Paul Erdős — Life and Work.- Paul Erdős Magic.- Part I Early Days.- Introduction.- Some of My Favorite Problems and Results.- 3 Encounters with Paul Erdős.- 4 Did Erdős Save Western Civilization?.- Integers Uniquely Represented by Certain Ternary Forms.- Did Erdős Save Western Civilization?.- Encounters with Paul Erdős.- On Cubic Graphs of Girth at Least Five.- Part II Number Theory.- Introduction.- Cross-disjoint Pairs of Clouds in the Interval Lattice.- Classical Results on Primitive and Recent Results on Cross-Primitive Sequences.- Dense Difference Sets and their Combinatorial Structure.- Integer Sets Containing No Solution to x+y=3z.- On Primes Recognizable in Deterministic Polynomial Time.- Ballot Numbers, Alternating Products, and the Erdős-Heilbronn Conjecture.- On Landau's Function g(n).- On Divisibility Properties on Sequences of Integers.- On Additive Representation Functions.- Arithmetical Properties of Polynomials.- Some Methods of Erdős Applied to Finite Arithmetic Progressions.- Sur La Non-Dérivabilité de Fonctions Périodiques Associées à Certaines Formules Sommatoires.- 1105: First Steps in a Mysterious Quest.- Part III Randomness and Applications.- Introduction.- Games, Randomness, and Algorithms.- The Origins of the Theory of Random Graphs.- An Upper bound for a Communication Game Related to Time-space Tradeoffs.- How Abelian is a Finite Group?.- One Small Size Approximation Models.- The Erdős Existence Argument.- Part IV Geometry.- Introduction.- Extension of Functional Equations.- Remarks on Penrose Tilings.- Distances in Convex Polygons.- Unexpected Applications of Polynomials in Combinatorics.- The Number of Homothetic Subsets.- On Lipschitz Mappings Onto a Square.- A Remark on Transversal Numbers.- In Praise of the Gram Matrix.- On Mutually Avoiding Sets.- Bibliography.