Paul Erdös was one of the most influential mathematicians of the twentieth century, whose work in number theory, combinatorics, set theory, analysis, and other branches of mathematics has determined the development of large areas of these fields. In 1999, a conference was organized to survey his work, his contributions to mathematics, and the far-reaching impact of his work on many branches of mathematics. On the 100th anniversary of his birth, this volume undertakes the almost impossible task to describe the ways in which problems raised by him and topics initiated by him (indeed, whole branches of mathematics) continue to flourish. Written by outstanding researchers in these areas, these papers include extensive surveys of classical results as well as of new developments.
Contents.- Preface.- Alon, N.: Paul Erdös and Probabilistic Reasoning.- Benjamini, I.: Euclidean vs. Graph Metric.- Bollobas, B. and Riordan, O.: The Phase Transition in the Erdös–Rényi Random Graph Process.- Bourgain, J.: Around the Sum-product Phenomenon.- Breuillard, E., Green, B. and Tao, T.: Small Doubling in Groups.- Diamond, H. G.: Erdös and Multiplicative Number Theory.- Füredi, Z. and Simonovits, M.: The History of Degenerate (Bipartite) Extremal Graph Problems.- Gowers, W. T.: Erdös and Arithmetic Progressions.- Graham, R. L.: Paul Erdös and Egyptian Fractions.- Györy, K.: Perfect Powers in Products with Consecutive Terms from Arithmetic Progressions.- Komjáth, P.: Erdös’s Work on Infinite Graphs.- Kunen, K.: The Impact of Paul Erd˝os on Set Theory.- Mauldin, R. D.: Some Problems and Ideas of Erdös in Analysis and Geometry.- Montgomery, H. L.: L2 Majorant Principles.- Nesetril, J.: A Combinatorial Classic – Sparse Graphs with High Chromatic Number.- Nguyen, H. H. and Vu, V. H.: Small Ball Probability, Inverse Theorems, and Applications.- Pach, J.: The Beginnings of Geometric Graph Theory.- Pintz, J.: Paul Erdös and the Difference of Primes.- Pollack, P. and Pomerance, C.: Paul Erdös and the Rise of Statistical Thinking in Elementary Number Theory.- Rödl, V. and Schacht, M.: Extremal Results in Random Graphs.-Schinzel, A.: Erdös’s Work on the Sum of Divisors Function and on Euler’s Function.- Shalev, A.: Some Results and Problems in the Theory of Word Maps.- Tenenbaum, G.: Some of Erdös’ Unconventional Problems in Number Theory, Thirty-four Years Later.- Totik, V.: Erdös on Polynomials.- Vertesi, P.: Paul Erdös and Interpolation: Problems, Results, New Developments.