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

  • JING ZHANG (a1)

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.

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[1]Cummings, J., Iterated forcing and elementary embeddings, Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), vol. 1, Springer, Dordrecht, 2010, pp. 775883.
[2]Devlin, D., Some partition theorems and ultrafilters on ω, Ph.D. thesis, Dartmouth College, 1979.
[3]Dobrinen, N. and Hathaway, D., Forcing and the Halpern-Läuchli theorem, submitted, 2017, arXiv:1706.08174.
[4]Dobrinen, N. and Hathaway, D., The Halpern-Läuchli theorem at a measurable cardinal, this Journal, vol. 82 (2017), no. 4, pp. 15601575.
[5]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. 237284.
[6]Foreman, M., Ideals and generic elementary embeddings, Handbook of Set Theory (Foreman, M. and Kanamori, A., editors), vol. 1, Springer, Dordrecht, 2010, pp. 8851147.
[7]Hajnal, A. and Komjáth, P., A strongly non-Ramsey order type. Combinatorica, vol. 17 (1997), no. 3, pp. 363367.
[8]Halpern, J. D. and Läuchli, H., A partition theorem. Transactions of the American Mathematical Society, vol. 124 (1966), pp. 360367.
[9]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. 83134.
[10]Kunen, K., Saturated ideals, this Journal, vol. 43 (1978), no. 1, pp. 6576.
[11]Laver, R., Products of infinitely many perfect trees. Journal of the London Mathematical Society (2), vol. 29 (1984), no. 3, pp. 385396.
[12]Mathias, A. R. D., Happy families. Annals of Mathematics Logic, vol. 12 (1977), no. 1, pp. 59111.
[13]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. 637668.
[14]Todorčević, S., Walks on Ordinals and Their Characteristics, Progress in Mathematics, vol. 263, Birkhäuser Verlag, Basel, 2007.
[15]Todorčević, S., Introduction to Ramsey Spaces, Annals of Mathematics Studies, vol. 174, Princeton University Press, Princeton, NJ, 2010.
[16]Todorčević, S. and Farah, I., Some Applications of the Method of Forcing, Yenisei Series in Pure and Applied Mathematics, Yenisei, Moscow; Lycée, Troitsk, 1995.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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