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A polarized partition relation for weakly compact cardinals using elementary substructures

Published online by Cambridge University Press:  12 March 2014

Albin L. Jones
Albin L. Jones*
Affiliation:
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KANSAS, LAWRENCE, KS 66045-2142, USA
*
2153 Oakdale Rd., Pasadena, MD 21122-5715, USA, E-mail: alj@mojumi.net, URL: http://www.mojumi.net/~alj
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Abstract

We show that if κ is a weakly compact cardinal, then

for any ordinals α < κ+ and μ < κ, and any finite ordinals m and n. This polarized partition relation represents the statement that for any partition

of κ × κ+ into m + μ pieces either there are A ∈ [κ]κ, B ∈ [κ]+]α and i < m with A × B ⊆ Ki or there are C ∈ [κ]κ, , and j < μ with C × D ⊆ Lj. Related results for measurable and almost measurable κ are also investigated. Our proofs of these relations involve the use of elementary substructures of set models of large fragments of ZFC.

Keywords

elementary substructures, measurable cardinals, normal ultrafilters, polarized partition relations, Ramsey theory, transfinite numbers, weakly compact cardinals
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1342 - 1352
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Baumgartner, J. and Hajnal, A., A proof (involving Martin's axiom) of a partition relation, Fundamenta Mathematicae, vol. 78 (1973), no. 3, pp. 193–203.CrossRefGoogle Scholar
[2]Baumgartner, James E. and Hajnal, Andras, Polarized partition relations, this Journal, vol. 66 (2001). no. 2, pp. 811–821.Google Scholar
[3]Baumgartner, James E., Hajnal, András, and Todorčević, Stevo, Extensions of the Erdős-Rado theorem, Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), Kluwer Academic Publishers, Dordrecht, 1993, pp. 1–17.Google Scholar
[4]Čudnovskiǐ, G. V., Combinatorial properties of compact cardinals, Infinite and finite sets (Colloquim, Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), vol. I, North-Holland, Amsterdam, 1975, pp. 289–306. Colloquia Mathematica Societatis János Bolyai, Vol. 10.Google Scholar
[5]Erdős, P. and Hajnal, A., Unsolved problems in set theory, Axiomatic set theory (Proceedings of Symposia on Pure Mathematics, vol. XIII, part I, University of California, Los Angeles, California, 1967), American Mathematical Society, Providence, R.I., 1971, pp. 17–48.Google Scholar
[6]Erdös, P. and Hajnal, A., Unsolved and solved problems in set theory, Proceedings of the Tarski Symposium (Proceedings of Symposia on Pure Mathematics, vol. XXV, University of California, Berkeley, California, 1971) (Providence, R.I.), American Mathematical Society, 1974, pp. 269–287.Google Scholar
[7]Erdős, P., Hajnal, A., and Rado, R., Partition relations for cardinal numbers, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16 (1965), pp. 93–196.CrossRefGoogle Scholar
[8]Erdős, P. and Rado, R., A partition calculus in set theory, Bulletin of the American Mathematical Society, vol. 62 (1956), pp. 427–489.CrossRefGoogle Scholar
[9]Hajnal, A., On some combinatorial problems involving large cardinals, Fundamenta Mathematicae, vol. 69 (1970), pp. 39–53.CrossRefGoogle Scholar
[10]Jech, Thomas, Set theory, second ed., Springer-Verlag, Berlin, 1997.CrossRefGoogle Scholar
[11]Jones, Albin L., A polarized partition relation using elementary substructures, this Journal, vol. 65 (2000), no. 4, pp. 1491–1498.Google Scholar
[12]Kanamori, A., Some combinatorics involving ultrafilters, Fundamenta Mathematicae, vol. 100 (1978), no. 2, pp. 145–155.CrossRefGoogle Scholar
[13]Milner, E. C., The use of elementary substructures in combinatorics, Discrete Mathematics, vol. 136 (1994), no. 1-3, pp. 243–252.CrossRefGoogle Scholar
[14]Wolfsdorf, Kurt, Der Beweis eines Satzes von G. Choodnovsky, Archiv für Mathematische Logik und Grundlagenforschung, vol. 20 (1980), no. 3-4, pp. 161–171.CrossRefGoogle Scholar
[15]Woodin, H., Unpublished works.Google Scholar