The Mathematics of Paul Erdös II

In 1992, when Paul Erdos was awarded a Doctor Honoris Causa by Charles University in Prague, a small conference was held, bringing together a distin­ guished group of researchers with interests spanning a variety of fields related to Erdos' own work. At that gathering, the idea occurred to several of us that it might be quite appropriate at this point in Erdos' career to solicit a col­ lection of articles illustrating various aspects of Erdos' mathematical life and work. The response to our solicitation was immediate and overwhelming, and these volumes are the result. Regarding the organization, we found it convenient to arrange the papers into six chapters, each mirroring Erdos' holistic approach to mathematics. Our goal was not merely a (random) collection of papers but rather a thor­ oughly edited volume composed in large part by articles explicitly solicited to illustrate interesting aspects of Erdos and his life and work. Each chap­ ter includes an introduction which often presents a sample of related Erdos' problems "in his own words". All these (sometimes lengthy) introductions were written jointly by editors. We wish to thank the nearly 70 contributors for their outstanding efforts (and their patience). In particular, we are grateful to Bela Bollobas for his extensive documentation of Paul Erdos' early years and mathematical high points (in the first part of this volume); our other authors are acknowledged in their respective chapters. We also want to thank A. Bondy, G. Hahn, I.

Contenu

of The Mathematics of Paul Erd?s II.- IV. Combinatorics and Graph Theory.- Problems in Graph Theory from Memphis.- Neighborly Families of Boxes and Bipartite Coverings.- Cycles and Paths in Triangle-Free Graphs.- Reconstruction Problems for Digraphs.- The Dimension of Random Graph Orders.- Hereditary and Monotone Properties of Graphs.- Properties of Graded Posets Preserved by Some Operations.- Intersection Representations of the Complete Bipartite Graph.- Reflections on a Problem of Erd?s and Hajnal.- The Chromatic Number of the Two-packing of a Forest.- On the Isolation of a Common Secret.- Some Remarks on the Cycle Plus Triangles Problem.- V. Ramsey and Extremal Theory.- Paul Erd?s' Influence on Extremal Graph Theory.- Ramsey Theory in the Work of Paul Erd?s.- Memories on Shadows and Shadows of Memories.- Applications of the Probabilistic Method to Partially Ordered Sets.- A Bound of the Cardinality of Families not Containing ?-Systems.- Arrangeability and Clique Subdivisions.- A Finite Partition Theorem with Double Exponential Bound.- VI. Geometry.- Extension of Functional Equations.- Remarks on Penrose Tilings.- Distances in Convex Polygons.- 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.- VII. Infinity.- The Random Graph.- Paul Erd?s' Set Theory.- A Few Remarks on a Conjecture of Erdos on the Infinite Version of Menger's Theorem.- On Order-Perfect Lattices.- The PCF Theorem Revisited.- Set Theory: Geometric and Real.- Paul Erd?s: The Master of Collaboration.- List of Publications of Paul Erdos.- Postscript.