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Combinatorica

, Volume 15, Issue 1, pp 85–104 | Cite as

On Erdős-Rado numbers

  • Hanno Lefmann
  • Vojtěch Rödl
Article

Abstract

In this paper new proofs of the Canonical Ramsey Theorem, which originally has been proved by Erdős and Rado, are given. These yield improvements over the known bounds for the arising Erdős-Rado numbersER(k; l), where the numbersER(k; l) are defined as the least positive integern such that for every partition of thek-element subsets of a totally orderedn-element setX into an arbitrary number of classes there exists anl-element subsetY ofX, such that the set ofk-element subsets ofY is partitioned canonically (in the sense of Erdős and Rado). In particular, it is shown that
$$2^{c1} .l^2 \leqslant ER(2;l) \leqslant 2^{c_2 .l^2 .\log l} $$
for every positive integerl≥3, wherec 1,c 2 are positive constants. Moreover, new bounds, lower and upper, for the numbersER(k; l) for arbitrary positive integersk, l are given.

Mathematics Subject Classification (1991)

05 A 99 

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References

  1. [1]
    M. Ajtai, J. Komlós, J. Pintz, J. Spencer, andE. Szemerédi: Extremal uncrowded hypergraphs,Journal of Combinatorial Theory Ser. A 32 (1982), 321–335.zbMATHCrossRefGoogle Scholar
  2. [2]
    N. Alon: On a conjecture of Erdős, T. Sós and Simonovits concerning anti-Ramsey theorems,Journal of Graph Theory 7 (1983), 91–94.zbMATHMathSciNetGoogle Scholar
  3. [3]
    N. Alon, H. Lefmann, andV. Rödl: On an anti-Ramsey type result,Colloquia Mathematica Societatis János Bolyai, 60. Sets, Graphs and Numbers, Budapest, 1991, 9–22.Google Scholar
  4. [4]
    L. Babai: An anti-Ramsey theorem,Graphs and Combinatorics,1 (1985), 23–28.zbMATHMathSciNetGoogle Scholar
  5. [5]
    J. Baumgartner: Canonical Partition Relations,The Journal of Symbolic Logic 40 (1975), 541–554.MathSciNetCrossRefGoogle Scholar
  6. [6]
    D. Duffus, H. Lefmann, andV. Rödl: Shift graphs and lower bounds on Ramsey numbersr k(l; r), to appear.Google Scholar
  7. [7]
    R. A. Duke, H. Lefmann, andV. Rödl: On uncrowded hypergraphs, 1992, to appear.Google Scholar
  8. [8]
    P. Erdős: Some remarks on the theory of graphs,Bull. Amer. Math. Soc. 53 (1947), 292–294.MathSciNetCrossRefGoogle Scholar
  9. [9]
    P. Erdős, andA. Hajnal: Ramsey-type theorems,Discrete Applied Mathematics 25 (1989), 37–52.MathSciNetCrossRefGoogle Scholar
  10. [10]
    P. Erdős, A. Hajnal, andR. Rado: Partition relations for cardinal numbers,Acta Math. Acad. Sci. Hung. 16 (1965), 93–196.CrossRefGoogle Scholar
  11. [11]
    P. Erdős, andL. Lovász: Problems and results on 3-chromatic hypergraphs and some related questions, in:Infinite and Finite Sets (A. Hajnal, R. Rado andV. T. Sós, eds.), North Holland, Amsterdam, 1975, 609–628.Google Scholar
  12. [12]
    P. Erdős, J. Nešetřil, andV. Rödl: On some problems related to partitions of edges in graphs, in:Graphs and other combinatorial topics, Proceedings of the third Czechoslovak Symposium on Graph Theory, ed. M. Fiedler, Teubner Texte in Mathematik vol. 59, Leipzig, 1983, 54–63.Google Scholar
  13. [13]
    P. Erdős, andR. Rado: A combinatorial theorem,Journal of the London Mathematical Society 25 (1950), 249–255.Google Scholar
  14. [14]
    P. Erdős, andR. Rado: Combinatorial theorems on classification of subsets of a given set,Proceedings London Mathematical Society 2 (1952), 417–439.Google Scholar
  15. [15]
    P. Erdős, V. T. Sós, andM. Simonovits: Anti-Ramsey Theorems, in:Infinite and Finite sets, Proceedings Kolloq. Keszthely, Hungary 1973, eds. A. Hajnal, R. Rado, V. T. Sós, vol. II, Colloq. Math. Soc. János Bolyai 10, Amsterdam, North Holland, 1975, 657–665.Google Scholar
  16. [16]
    P. Erdős, andJ. Spencer:Probabilistic methods in combinatorics, Academic Press, New York, 1974.Google Scholar
  17. [17]
    P. Erdős, andE. Szemerédi: On a Ramsey type theorem,Periodica Mathematica Hungarica 2 (1972), 1–4.Google Scholar
  18. [18]
    R. L. Graham, B. L. Rothschild, andJ. H. Spencer:Ramsey Theory, 2nd edition, Wiley-Interscience, New York, 1989.Google Scholar
  19. [19]
    H. Lefmann: A note on Ramsey numbers,Studia Scientiarum Mathematicarum Hungarica 22 (1987), 445–446.zbMATHMathSciNetGoogle Scholar
  20. [20]
    H. Lefmann, andV. Rödl: On canonical Ramsey numbers for coloring three-element sets, in:Finite and Infinite Combinatorics in Sets and Logic, (eds.: N. W. Sauer, R. E. Woodrow, B. Sands), Kluwer 1993, 237–247.Google Scholar
  21. [21]
    H. Lefmann, andV. Rödl: On canonical Ramsey numbers for complete graphs versus paths,Journal of Combinatorial Theory Ser. B 58 (1993), 1–13.zbMATHCrossRefGoogle Scholar
  22. [22]
    R. Rado: Anti-Ramsey Theorems, in:Finite and Infinite Sets, eds. Hajnal, Rado, Sós, Coll. Math. Soc. János Bolyai, North Holland, 1975, 1159–1168.Google Scholar
  23. [23]
    R. Rado: Note on canonical partitions,Bulletin London Mathematical Society 18 (1986), 123–126.zbMATHMathSciNetGoogle Scholar
  24. [24]
    F. P. Ramsey: On a problem of formal logic,Proceedings London Mathematical Society 30 (1930), 264–286.Google Scholar
  25. [25]
    M. Simonovits, andV. T. Sós: On restricted colorings ofK n,Combinatorica 4 (1984), 101–110.zbMATHMathSciNetGoogle Scholar
  26. [26]
    J. Spencer: Asymptotic lower bounds for Ramsey functions,Discrete Mathematics 20 (1977), 69–77.MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó 1995

Authors and Affiliations

  • Hanno Lefmann
    • 1
  • Vojtěch Rödl
    • 2
  1. 1.Lehrstuhl Informatik IIUniversität DortmundDortmundGermany
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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