Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T22:31:30.007Z Has data issue: false hasContentIssue false
  • English
  • Français

A polarized partition relation using elementary substructures

Published online by Cambridge University Press:  12 March 2014

Albin L. Jones
Albin L. Jones*
Affiliation:
Department of Mathematics, Kenyon College, Gambier, OH 43022, USA, E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Working in ZFC, we show that for any infinite cardinal k and ordinal y < (2<k)+ the polarized partition relation

holds. Our proof of this relation involves the use of elementary substructures of set models of large fragments of ZFC.

Keywords

elementary substructurespolarized partition relationsRamsey theorytransfinite numbers
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 4 , December 2000 , pp. 1491 - 1498
Copyright
Copyright © Association for Symbolic Logic 2000

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

REFERENCES

[1]Baumgartner, J., Hajnal, A., and Todorčević, S., Extensions of the Erdős-Rado theorem, Finite and infinite combinatorics in sets and logic (Dordrecht), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 411, Kluwer Academic Publications, 1993, pp. 1–17.Google Scholar
[2]Erdős, P. and Hajnal, A., Unsolved and solved problems in set theory, Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics, vol. 25, 1971, pp. 269–287.CrossRefGoogle Scholar
[3]Erdős, P. and Hajnal, A., Unsolved problems in set theory, Part 1 (Providence, RI), Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, 1971, pp. 17–48.Google Scholar
[4]Erdős, P., Hajnal, A., Mate, A., and Rado, R., Combinatorial set theory: Partition relations for cardinals, Studies in Logic and the Foundations of Mathematics, vol. 106, North-Holland, Amsterdam, 1984.Google Scholar
[5]Erdős, P., Hajnal, A., and Rado, R., Partition relations for cardinal numbers, Acta Mathhematica Academiae Scientia Hungarica, vol. 16 (1965), pp. 93–196.CrossRefGoogle Scholar
[6] P. Erdős and Rado, R., A partition calculus in set theory, Bulletin of the American Mathematical Society, vol. 62 (1956), pp. 427–489.Google Scholar
[7]Jech, T., Set theory, first ed., Pure and Applied Mathematics, vol. 79, Academic Press, 1978.Google Scholar
[8]Milner, E. C., The use of elementary substructures in combinatorics, Discrete Mathematics, vol. 136 (1994), pp. 243–252.CrossRefGoogle Scholar
[9]Prikry, K., On aproblem of Erdos, Hajnal, and Rado, Discrete Mathematics, vol. 2 (1972), pp. 51–59.CrossRefGoogle Scholar