### 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 < 2
^{n-1} (proof: color randomly!). We give an argument of Srinivasan and Radhakrishnan ([RS]) that extends this to m < c2
^{n} √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 language | English (US) |
---|---|

Title of host publication | FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science - 25th International Conference, Proceedings |

Pages | 106 |

Number of pages | 1 |

Volume | 3821 LNCS |

State | Published - 2005 |

Event | 25th International Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2005 - Hyderabad, India Duration: Dec 15 2005 → Dec 18 2005 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 3821 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 25th International Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2005 |
---|---|

Country | India |

City | Hyderabad |

Period | 12/15/05 → 12/18/05 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

*FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science - 25th International Conference, Proceedings*(Vol. 3821 LNCS, pp. 106). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 3821 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science - 25th International Conference, Proceedings.*vol. 3821 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 3821 LNCS, pp. 106, 25th International Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2005, Hyderabad, India, 12/15/05.

}

TY - GEN

T1 - Erdös magic

AU - Spencer, Joel

PY - 2005

Y1 - 2005

N2 - 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 < 2 n-1 (proof: color randomly!). We give an argument of Srinivasan and Radhakrishnan ([RS]) that extends this to m < c2 n √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?

AB - 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 < 2 n-1 (proof: color randomly!). We give an argument of Srinivasan and Radhakrishnan ([RS]) that extends this to m < c2 n √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?

UR - http://www.scopus.com/inward/record.url?scp=33744921900&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33744921900&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33744921900

SN - 3540304959

SN - 9783540304951

VL - 3821 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 106

BT - FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science - 25th International Conference, Proceedings

ER -