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An order property of partition cardinals

Published online by Cambridge University Press:  17 April 2009

N. H. Williams
N. H. Williams
Affiliation:
University of the Witwatersrand, Johannesburg, South Africa.
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Abstract

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This note studies cardinal numbers κ which have a partition property which amounts to the following. Let ν be a cardinal, η an ordinal limit number and m a positive integer. Let the m-length sequences of finite subsets of κ be partitioned into ν parts. Then there is a sequence H1, … Hm of subsets of κ, each having order type η, such that for each choice of non-zero numbers n1, …, nm there is some class of the partition inside which fall all sequences having in their i-th place (for i = 1, …, m) a subset of Hi which contains exactly ni elements. The case when m = 1 is thus seen to be the well known property κ . The most interesting results obtained relate to the ordering of the least cardinals with the appropriate properties as m and η vary.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 3 , Issue 2 , October 1970 , pp. 171 - 178
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Erdös, P., Hajnal, A. and Rado, R., “Partition relations for cardinal numbers”, Acta. Math. Acad. Sci. Hungar. 16 (1965), 93–196.CrossRefGoogle Scholar
[2]Morely, Michael, “Partitions and models”, Proceedings of the Summer School in Logic, Leeds, 1967. (Lecture notes in mathematies, 70, 109–158; Springer-Verlag, Berlin, Heidelberg, New York, 1968).CrossRefGoogle Scholar
[3]Rowbottom, Frederick, “Large cardinals and small constructible sets”, Ph.D. Thesis, University of Wisconsin, Madison, 1964.Google Scholar
[4]Silver, Jack Howard, “Some applications of model theory in set theory”, Ph.D. Thesis, University of California, Berkeley, 1966.Google Scholar
[5]Williams, Neil Hale, “Cardinal numbers with partition properties”, Ph.D. Thesis, Australian National University, Canberra, 1969.Google Scholar