Erdös magic

Joel Spencer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The Probabilistic Method ([AS]) is a lasting legacy of the late Paul Erdös. We give two examples - both problems first formulated by Erdös in the 1960s with new results in the last decade and both with substantial open questions. Further in both examples we take a Computer Science vantagepoint, creating a probabilistic algorithm to create the object (coloring, packing, respectively) and showing that with positive probability the created object has the desired properties. - Given m sets each of size n (with an arbitrary intersection pattern) we want to color the underlying vertices Red and Blue so that no set is monochromatic. Erdös showed this may always be done if m < 2n-1 (proof: color randomly!). We give an argument of Srinivasan and Radhakrishnan ([RS]) that extends this to m < c2n √n/ ln n. One first colors randomly and then recolors the blemishes with a clever random sequential algorithm. - In a universe of size N we have a family of sets, each of size k, such that each vertex is in D sets and any two vertices have only o(D) common sets. Asymptotics are for fixed k with N, D → ∞. We want an asymptotic packing, a subfamily of ∼ N/k disjoint sets. Erdös and Hanani conjectured such a packing exists (in an important special case of asymptotic designs) and this conjecture was shown by Rödl. We give a simple proof of the author ([S]) that analyzes the random greedy algorithm. Paul Erdös was a unique figure, an inspirational figure to countless mathematicians, including the author. Why did his view of mathematics resonate so powerfully? What was it that drew so many of us into his circle? Why do we love to tell Erdös stories? What was the magic of the man we all knew as Uncle Paul?

Original languageEnglish (US)
Title of host publicationFSTTCS 2005
Subtitle of host publicationFoundations of Software Technology and Theoretical Computer Science - 25th International Conference, Proceedings
PublisherSpringer Verlag
Number of pages1
ISBN (Print)3540304959, 9783540304951
DOIs
StatePublished - 2005
Event25th International Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2005 - Hyderabad, India
Duration: Dec 15 2005Dec 18 2005

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume3821 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other25th International Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2005
CountryIndia
CityHyderabad
Period12/15/0512/18/05

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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