Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-25T07:06:30.201Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong partition properties for infinite cardinals

Published online by Cambridge University Press:  12 March 2014

E. M. Kleinberg
E. M. Kleinberg*
Affiliation:
The Rockefeller University Massachusetts Institute of Technology
Get access Rights & Permissions [Opens in a new window]

Extract

The notion of a “partition relation”, as it has been studied in the context of set theory for the past several years, was inspired by the following theorem of F. P. Ramsey [14]:

Theorem 0.1. Let n be a positive integer and let {A, B} be a partition of those subsets of the nonnegative integers containing exactly n elements. Then there exists an infinite subset x of the nonnegative integers all of whose n-element subsets are contained in only one of A or B. (Any such set x is said to be “homogeneous” for the partition.)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 35 , Issue 3 , September 1970 , pp. 410 - 428
Copyright
Copyright © Association for Symbolic Logic 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Erdös, P. and Hajnal, A., On the structure of set mappings, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 9 (1958), pp. 111–131.CrossRefGoogle Scholar
[2]Erdös, P. and Hajnal, A., Some remarks concerning our paper “ On the structure of set mappings”, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 13 (1962), pp. 223–226.CrossRefGoogle Scholar
[3]Erdös, P., Hajnal, A. and Rado, R., Partition relations for cardinal numbers, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16 (1965), pp. 93–196.CrossRefGoogle Scholar
[4]Erdös, P. and Rado, R., Combinatorial theorems on classification of subsets of a given set, Proceedings of the London Mathematical Society (3), vol. 2 (1952), pp. 417–439.CrossRefGoogle Scholar
[5]Erdös, P. and Rado, R., A partition calculus in set theory, Bulletin of the American Mathematical Society, vol. 62 (1956), pp. 427–189.CrossRefGoogle Scholar
[6]Galvin, F., A generalization of Ramsey's theorem, Notices of the American Mathematical Society, vol. 15 (1968), p. 548.Google Scholar
[7]Jensen, R., Ramsey cardinals and the general continuum hypothesis, Notices of the American Mathematical Society, vol. 14 (1967), p. 253.Google Scholar
[8]Kleinberg, E. M., Strong partition properties, Notices of the American Mathematical Society, vol. 16 (1969), p. 579.Google Scholar
[9]Kleinberg, E. M., The independence of Ramsey's theorem, this Journal, vol. 34 (1969), pp. 205–206.Google Scholar
[10]Kleinberg, E. M., Somewhat homogeneous sets, Notices of the American Mathematical Society, vol. 16 (1969), p. 840.Google Scholar
[11]Martin, D. A., Measurable cardinals and analytic games, Fundamenta mathematicae (to appear).Google Scholar
[12]Mathias, A. R. D., Doctoral dissertation, Peterhouse, Cambridge Univ., Cambridge, England. See A. R. D. Mathias, On a generalization of Ramsey's theorem, Notices of the American Mathematical Society, vol. 15 (1968), p. 931.Google Scholar
[13]Morley, M., Categoricity in power, Doctoral dissertation, University of Chicago, Chicago, Ill., 1962.Google Scholar
[14]Ramsey, F. P., On a problem of formal logic, Proceedings of the London Mathematical Society (2), vol. 30 (1930), pp. 264–286.CrossRefGoogle Scholar
[15]Rowbottom, F., Large cardinals and constructible sets, Doctoral dissertation, University of Wisconsin, Madison, Wis., 1964.Google Scholar
[16]Silver, J., Some applications of model theory in set theory, Doctoral dissertation, University of California, Berkeley, Calif., 1966.Google Scholar