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A TAIL CONE VERSION OF THE HALPERN–LÄUCHLI THEOREM AT A LARGE CARDINAL

Published online by Cambridge University Press:  08 April 2019

JING ZHANG
JING ZHANG*
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PENNSYLVANIA 15213, USA E-mail: jingzhang@cmu.edu
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Abstract

The classical Halpern–Läuchli theorem states that for any finite coloring of a finite product of finitely branching perfect trees of height ω, there exist strong subtrees sharing the same level set such that tuples in the product of the strong subtrees consisting of elements lying on the same level get the same color. Relative to large cardinals, we establish the consistency of a tail cone version of the Halpern–Läuchli theorem at a large cardinal (see Theorem 3.1), which, roughly speaking, deals with many colorings simultaneously and diagonally. Among other applications, we generalize a polarized partition relation on rational numbers due to Laver and Galvin to one on linear orders of larger saturation.

Keywords

Primary 03E0203E3503E55Halpern-Läuchli theoremRamsey theorytreesforcinglarge cardinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 2 , June 2019 , pp. 473 - 496
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

REFERENCES

Cummings, J., Iterated forcing and elementary embeddings, Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), vol. 1, Springer, Dordrecht, 2010, pp. 775–883.Google Scholar
Devlin, D., Some partition theorems and ultrafilters on ω, Ph.D. thesis, Dartmouth College, 1979.Google Scholar
Dobrinen, N. and Hathaway, D., Forcing and the Halpern-Läuchli theorem, submitted, 2017, arXiv:1706.08174.Google Scholar
Dobrinen, N. and Hathaway, D., The Halpern-Läuchli theorem at a measurable cardinal, this Journal, vol. 82 (2017), no. 4, pp. 1560–1575.Google Scholar
Džamonja, M., Larson, J. A., and Mitchell, W. J., A partition theorem for a large dense linear order. Israel Journal of Mathematics, vol. 171 (2009), pp. 237–284.Google Scholar
Foreman, M., Ideals and generic elementary embeddings, Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), vol. 1, Springer, Dordrecht, 2010, pp. 885–1147.Google Scholar
Hajnal, A. and Komjáth, P., A strongly non-Ramsey order type. Combinatorica, vol. 17 (1997), no. 3, pp. 363–367.Google Scholar
Halpern, J. D. and Läuchli, H., A partition theorem. Transactions of the American Mathematical Society, vol. 124 (1966), pp. 360–367.Google Scholar
Halpern, J. D. and Lévy, A., The Boolean prime ideal theorem does not imply the axiom of choice, Axiomatic Set Theory (Proceedings of Symposia in Pure Mathematics, Vol. XIII, Part I, University of California, Los Angeles, California, 1967) (Scott, D. S., editor), American Mathematical Society, Providence, RI, 1971, pp. 83–134.Google Scholar
Kunen, K., Saturated ideals, this Journal, vol. 43 (1978), no. 1, pp. 65–76.Google Scholar
Laver, R., Products of infinitely many perfect trees. Journal of the London Mathematical Society (2), vol. 29 (1984), no. 3, pp. 385–396.Google Scholar
Mathias, A. R. D., Happy families. Annals of Mathematics Logic, vol. 12 (1977), no. 1, pp. 59–111.Google Scholar
Shelah, S., Strong partition relations below the power set: Consistency; was Sierpiński right? II, Sets, Graphs and Numbers (Budapest, 1991) (Hálasz, G., editor), Colloquia Mathematica Societatis János Bolyai, vol. 60, North-Holland, Amsterdam, 1992, pp. 637–668.Google Scholar
Todorčević, S., Walks on Ordinals and Their Characteristics, Progress in Mathematics, vol. 263, Birkhäuser Verlag, Basel, 2007.Google Scholar
Todorčević, S., Introduction to Ramsey Spaces, Annals of Mathematics Studies, vol. 174, Princeton University Press, Princeton, NJ, 2010.Google Scholar
Todorčević, S. and Farah, I., Some Applications of the Method of Forcing, Yenisei Series in Pure and Applied Mathematics, Yenisei, Moscow; Lycée, Troitsk, 1995.Google Scholar