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
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.
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M. Ajtai, J. Komlós, J. Pintz, J. Spencer, andE. Szemerédi: Extremal uncrowded hypergraphs,Journal of Combinatorial Theory Ser. A 32 (1982), 321–335.
N. Alon: On a conjecture of Erdős, T. Sós and Simonovits concerning anti-Ramsey theorems,Journal of Graph Theory 7 (1983), 91–94.
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.
L. Babai: An anti-Ramsey theorem,Graphs and Combinatorics,1 (1985), 23–28.
J. Baumgartner: Canonical Partition Relations,The Journal of Symbolic Logic 40 (1975), 541–554.
D. Duffus, H. Lefmann, andV. Rödl: Shift graphs and lower bounds on Ramsey numbersr k(l; r), to appear.
R. A. Duke, H. Lefmann, andV. Rödl: On uncrowded hypergraphs, 1992, to appear.
P. Erdős: Some remarks on the theory of graphs,Bull. Amer. Math. Soc. 53 (1947), 292–294.
P. Erdős, andA. Hajnal: Ramsey-type theorems,Discrete Applied Mathematics 25 (1989), 37–52.
P. Erdős, A. Hajnal, andR. Rado: Partition relations for cardinal numbers,Acta Math. Acad. Sci. Hung. 16 (1965), 93–196.
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.
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.
P. Erdős, andR. Rado: A combinatorial theorem,Journal of the London Mathematical Society 25 (1950), 249–255.
P. Erdős, andR. Rado: Combinatorial theorems on classification of subsets of a given set,Proceedings London Mathematical Society 2 (1952), 417–439.
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.
P. Erdős, andJ. Spencer:Probabilistic methods in combinatorics, Academic Press, New York, 1974.
P. Erdős, andE. Szemerédi: On a Ramsey type theorem,Periodica Mathematica Hungarica 2 (1972), 1–4.
R. L. Graham, B. L. Rothschild, andJ. H. Spencer:Ramsey Theory, 2nd edition, Wiley-Interscience, New York, 1989.
H. Lefmann: A note on Ramsey numbers,Studia Scientiarum Mathematicarum Hungarica 22 (1987), 445–446.
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.
H. Lefmann, andV. Rödl: On canonical Ramsey numbers for complete graphs versus paths,Journal of Combinatorial Theory Ser. B 58 (1993), 1–13.
R. Rado: Anti-Ramsey Theorems, in:Finite and Infinite Sets, eds. Hajnal, Rado, Sós, Coll. Math. Soc. János Bolyai, North Holland, 1975, 1159–1168.
R. Rado: Note on canonical partitions,Bulletin London Mathematical Society 18 (1986), 123–126.
F. P. Ramsey: On a problem of formal logic,Proceedings London Mathematical Society 30 (1930), 264–286.
M. Simonovits, andV. T. Sós: On restricted colorings ofK n,Combinatorica 4 (1984), 101–110.
J. Spencer: Asymptotic lower bounds for Ramsey functions,Discrete Mathematics 20 (1977), 69–77.
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Supported by NSF Grant DUS-9011850.
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Lefmann, H., Rödl, V. On Erdős-Rado numbers. Combinatorica 15, 85–104 (1995). https://doi.org/10.1007/BF01294461
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DOI: https://doi.org/10.1007/BF01294461