On the chromatic number of set systems

  1. Alexandr Kostochka1,*,
  2. Dhruv Mubayi2,
  3. Vojtěch Rödl3,
  4. Prasad Tetali2

Article first published online: 17 AUG 2001

DOI: 10.1002/rsa.1020

Issue

Random Structures & Algorithms

Random Structures & Algorithms

Volume 19, Issue 2, pages 87–98, September 2001

Additional Information

How to Cite

Kostochka, A., Mubayi, D., Rödl, V. and Tetali, P. (2001), On the chromatic number of set systems. Random Structures & Algorithms, 19: 87–98. doi: 10.1002/rsa.1020

Author Information

  1. 1

    Department of Mathematics, University of Illinois, Urbana, Illinois 61801 and Institute of Mathematics, Novosibirsk, 630090, Russia

  2. 2

    School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

  3. 3

    Department of Mathematics, Emory University, Atlanta, Georgia 30322

*Department of Mathematics, University of Illinois, Urbana, Illinois 61801 and Institute of Mathematics, Novosibirsk, 630090, Russia

Publication History

  1. Issue published online: 17 AUG 2001
  2. Article first published online: 17 AUG 2001
  3. Manuscript Accepted: 6 MAR 2001
  4. Manuscript Received: 24 NOV 2000

Funded by

  • RFBR. Grant Numbers: 99-01-00581, 00-01-00916
  • NSF. Grant Numbers: DMS9970325, DMS0071261, DMS9800351

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Keywords:

  • extremal set theory;
  • uniform hypergraph;
  • property B

Abstract

An (r, l)-system is an r-uniform hypergraph in which every set of l vertices lies in at most one edge. Let mk(r, l) be the minimum number of edges in an (r, l)-system that is not k-colorable. Using probabilistic techniques, we prove that

  • equation image

where br, l is explicitly defined and ar, l is sufficiently small. We also give a different argument proving (for even k)

  • equation image

where a′r, l=(r−l+1)/r(2r−1re)−l/(l−1).

Our results complement earlier results of Erdős and Lovász [10] who mainly focused on the case l=2, k fixed, and r large. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 87–98, 2001

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