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Proof of a conjecture of Galvin

Part of: Extremal combinatorics Graph theory Set theory Spaces with richer structures

Published online by Cambridge University Press:  21 December 2020

Dilip Raghavan  and
Stevo Todorcevic
Dilip Raghavan
Affiliation:
Department of Mathematics, National University of Singapore, Singapore119076; E-mail: dilip.raghavan@protonmail.com
Stevo Todorcevic
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, Canada, M5S 2E4; E-mail: stevo@math.toronto.edu Institut de Mathématique de Jussieu, UMR 7586, Case 247, 4 place Jussieu, 75252Paris Cedex, France; E-mail: todorcevic@math.jussieu.fr

Abstract

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We prove that if the set of unordered pairs of real numbers is coloured by finitely many colours, there is a set of reals homeomorphic to the rationals whose pairs have at most two colours. Our proof uses large cardinals and verifies a conjecture of Galvin from the 1970s. We extend this result to an essentially optimal class of topological spaces in place of the reals.

MSC classification

Primary: 03E02: Partition relations
Secondary: 05D10: Ramsey theory 03E55: Large cardinals 05C55: Generalized Ramsey theory 54E40: Special maps on metric spaces
Type
Foundations
Information
Forum of Mathematics, Pi , Volume 8 , 2020 , e15
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Baumgartner, J. E., ‘Partition relations for countable topological spaces’, J. Combin. Theory Ser. A 43(2) (1986), 178–195.CrossRefGoogle Scholar
Borel, E., ‘Sur la classification des ensembles de mesure nulle’, Bull. Soc. Math. France 47 (1919), 97–125.CrossRefGoogle Scholar
Devlin, D. C., Some Partition Theorems and Ultrafilters on $\omega$, Ph.D. thesis, Dartmouth College, Ann Arbor, MI, 1980.Google Scholar
Ehrenfeucht, A. and Mostowski, A., ‘Models of axiomatic theories admitting automorphisms’, Fund. Math. 43 (1956), 50–68.CrossRefGoogle Scholar
Erdős, P. and Rado, R., ‘Combinatorial theorems on classifications of subsets of a given set’, Proc. Lond. Math. Soc. (3) 2 (1952), 417–439.CrossRefGoogle Scholar
Fleissner, W. G., ‘Left separated spaces with point-countable bases’, Trans. Amer. Math. Soc. 294(2) (1986), 665–677.CrossRefGoogle Scholar
Foreman, M., Ideals and Generic Elementary Embeddings, Handbook of Set Theory, Vol. 2 (Springer, Dordrecht, 2010), 885–1147.CrossRefGoogle Scholar
Foreman, M., Magidor, M. and Shelah, S., ‘Martin’s maximum, saturated ideals, and nonregular ultrafilters, I’, Ann. of Math. (2) 127(1) (1988), 1–47.CrossRefGoogle Scholar
Galvin, F., Letter to R. Laver (March 19, 1970).Google Scholar
Gerlits, J. and Szentmiklóssy, Z., ‘A Ramsey-type topological theorem’, Topology Appl. 125(2) (2002), 343–355.CrossRefGoogle Scholar
Haydon, R., Some problems about scattered spaces, Séminaire d’Initiation à~l’Analyse, Publ. Math. Univ. Pierre et Marie Curie, vol. 95, Univ. Paris VI, Paris, 1989/1990, pp. Exp. No. 9, 10.Google Scholar
Jech, T., Set Theory, third millennium edn., revised and expanded, Springer Monogr. Math. (Springer-Verlag, Berlin, 2003).Google Scholar
Jensen, R. and Steel, J., ‘$\!\!K$ without the measurable’, J. Symb. Log. 78(3) (2013), 708–734.CrossRefGoogle Scholar
Kanamori, A., The Higher Infinite, second ed., Springer Monogr. Math. (Springer-Verlag, Berlin, 2009), Large cardinals in set theory from their beginnings, MathsciNet, Paperback reprint of the 2003 edition.Google Scholar
Kechris, A. S., Pestov, V. G. and Todorcevic, S., ‘Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups’, Geom. Funct. Anal. 15(1) (2005), 106–189.CrossRefGoogle Scholar
Larson, P. B., The Stationary Tower, Univ. Lecture Ser., 32 (American Mathematical Society, Providence, RI, 2004), Notes on a course by W. Hugh Woodin.Google Scholar
Laver, R., ‘On the consistency of Borel’s conjecture’, Acta Math. 137(3-4) (1976), 151–169.CrossRefGoogle Scholar
Magidor, M., ‘How large is the first strongly compact cardinal? or A study on identity crises’, Ann. Math. Logic 10(1) (1976), 33–57.CrossRefGoogle Scholar
Raghavan, D. and Todorcevic, S., ‘Galvin’s problem in higher dimension’, In preparation.Google Scholar
Ramsey, F. P., ‘On a problem of formal logic’, Proc. Lond. Math. Soc. (2) 30(4) (1929), 264–286.Google Scholar
Shelah, S., ‘Strong partition relations below the power set: Consistency; was Sierpiński right? II’, in Sets, Graphs and Numbers (Budapest, 1991), Colloquia Mathematica Societatis János Bolyai, 60 (North-Holland, Amsterdam, 1992), 637–668.Google Scholar
Shelah, S., ‘Was Sierpiński right? IV’, J. Symb. Log. 65(3) (2000), 1031–1054.CrossRefGoogle Scholar
Sierpiński, W., ‘Sur une problème de la théorie des relations’, Ann. Sc. Norm. Super. Pisa (2) 2 (1933), 239–242.Google Scholar
Todorcevic, S., ‘Partition relations for partially ordered sets’, Acta Math. 155(1-2) (1985), 1–25.CrossRefGoogle Scholar
Todorcevic, S., ‘Partitioning pairs of countable ordinals’, Acta Math. 159(3-4) (1987), 261–294.CrossRefGoogle Scholar
Todorcevic, S., ‘A partition property of spaces with point-countable bases’, Unpublished notes (June 1996).Google Scholar
Todorcevic, S., ‘A dichotomy for P-ideals of countable sets’, Fund. Math. 166(3) (2000), 251–267.CrossRefGoogle Scholar
Todorcevic, S., ‘Universally meager sets and principles of generic continuity and selection in Banach spaces’, Adv. Math. 208(1) (2007), 274–298.CrossRefGoogle Scholar
Todorcevic, S., Introduction to Ramsey Spaces, Ann. of Math. Stud., 174 (Princeton University Press, Princeton, NJ, 2010).Google Scholar
Todorcevic, S. and Weiss, W., ‘Partitioning metric spaces’, Unpublished manuscript (September 1995).Google Scholar
Trang, N., ‘$\mathrm{PFA}$ and guessing models’, Israel J. Math. 215(2) (2016), 607–667.CrossRefGoogle Scholar
Woodin, W. H., ‘Supercompact cardinals, sets of reals, and weakly homogeneous trees’, Proc. Natl. Acad. Sci. USA 85(18) (1988), 6587–6591.CrossRefGoogle ScholarPubMed
Woodin, W. H., The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, de Gruyter Series in Logic and its Applications, 1 (Walter de Gruyter & Co., Berlin, 1999).Google Scholar