A Quasi Ramsey Theorem

  • Hans Jürgen Prömel
Chapter

Abstract

The basic problem of (combinatorial) discrepancy theory is how to color a set with two colors as uniformly as possible with respect to a given family of subsets. The aim is to achieve that each of the two colors meets each subset under consideration in approximately the same number of elements. From the finite Ramsey theorem (cf. Corollary 7.2) we know already that if the set of all 2-subsets of n is 2-colored, and the family of all -subsets for some \(\ell< \frac{1} {2}\log n\) is considered, the situation is as bad as possible: for any 2-coloring we will find a monochromatic -set. As gets larger one can color more uniformly though one still has the preponderance phenomenon.

References

  1. Beck, J., Sós, V.T.: Discrepancy theory. In: Handbook of Combinatorics, vols. 1, 2, pp. 1405–1446. Elsevier, Amsterdam (1995)Google Scholar
  2. Chazelle, B.: The Discrepancy Method. Cambridge University Press, Cambridge (2000)CrossRefMATHGoogle Scholar
  3. Chernoff, H.: A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23, 493–507 (1952)MathSciNetCrossRefMATHGoogle Scholar
  4. Erdős, P.: On a lemma of Littlewood and Offord. Bull. Am. Math. Soc. 51, 898–902 (1945)CrossRefGoogle Scholar
  5. Erdős, P.: On combinatorial questions connected with a theorem of Ramsey and van der Waerden (in hungarian). Mat. Lapok 14, 29–37 (1963)MathSciNetGoogle Scholar
  6. Erdős, P., Rado, R.: A combinatorial theorem. J. Lond. Math. Soc. 25, 249–255 (1950)CrossRefGoogle Scholar
  7. Erdős, P., Spencer, J.: Imbalances in k-colorations. Networks 1, 379–385 (1971/1972)Google Scholar
  8. Littlewood, J.E., Offord, A.C.: On the number of real roots of a random algebraic equation. III. Rec. Math. [Mat. Sbornik] N.S. 12, 277–286 (1943)Google Scholar
  9. Sós, V.T.: Irregularities of partitions: Ramsey theory, uniform distribution. In: Surveys in Combinatorics (Southampton, 1983), pp. 201–246. Cambridge University Press, Cambridge (1983)Google Scholar
  10. Spencer, J.: Probabilistic methods. Graphs Comb. 1, 357–382 (1985)CrossRefMATHGoogle Scholar
  11. Sperner, E.: Ein Satz über Untermengen einer endlichen Menge. Mathematische Zeitschrift 27, 544–548 (1928)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Hans Jürgen Prömel
    • 1
  1. 1.Technische Universität DarmstadtDarmstadtGermany

Personalised recommendations