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Weak compactness and square bracket partition relations

Published online by Cambridge University Press:  12 March 2014

E. M. Kleinberg  and
R. A. Shore
E. M. Kleinberg
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
R. A. Shore
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
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Extract

Although there are many characterizations of weakly compact cardinals (e.g. in terms of indescnbability and tree properties as well as compactness) the most interesting set-theoretic (combinatorial) one is in terms of partition relations. To be more precise we define for κ and α cardinals and n an integer the partition relation of Erdös, Hajnal and Rado [2] as follows:

For every function F: [κ]n→ α (called a partition of [κ]n, the n-element subsets of κ, into α pieces), there exists a set C⊆ κ (called homogeneous for F) such that card C = κ and F″[C]n≠ α, i.e. some element of the range is omitted when F is restricted to the n-element subsets of C. It is the simplest (nontrivial) of these relations, i.e. , that is the well-known equivalent of weak compactness.1

Two directions of inquiry immediately suggest themselves when weak compactness is described in terms of these partition relations: (a) Trying to strengthen the relation by increasing the superscript—e.g., —and (b) trying to weaken the relation by increasing the subscript—e.g., . As it turns out, the strengthening to is only illusory for using the equivalence of to the tree property one quickly sees that implies (and so is equivalent to) for every n. Thus is the strongest of these partition relations. The second question seems much more difficult.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 37 , Issue 4 , December 1972 , pp. 673 - 676
Copyright
Copyright © Association for Symbolic Logic 1972

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References

BIBLIOGRAPHY

[1]Blass, A., Weak partition relations, Proceedings of the American Mathematical Society, vol. 35 (1972).CrossRefGoogle Scholar
[2]Erdös, P., Hajnal, A. and Rado, R., Partition relations for cardinals, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16 (1965), pp. 93–196.CrossRefGoogle Scholar
[3]Galvin, F. and Shelah, S., Negation of partition relations without CH, Notices of the American Mathematical Society, vol. 18 (1971), p. 666.Google Scholar
[4]Kleinberg, E. M., Somewhat homogeneous sets. III, Notices of the American Mathematical Society, vol. 17 (1970), p. 296.Google Scholar
[5]Shore, R. A., Square bracket partition relations in L (to appear).Google Scholar
[6]Silver, J., Some applications of model theory in set theory. Doctoral Dissertation, University of California, Berkeley, Calif., 1966.Google Scholar