Geodesic
Proof of a conjecture of Galvin
Forum of Mathematics, Pi, Tome 8 (2020)

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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. 
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Dilip Raghavan; Stevo Todorcevic. Proof of a conjecture of Galvin. Forum of Mathematics, Pi, Tome 8 (2020). doi: 10.1017/fmp.2020.12

[1] Baumgartner, J. E., ‘Partition relations for countable topological spaces’, J. Combin. Theory Ser. A 43(2) (1986), 178–195.Google Scholar | DOI

[2] Borel, E., ‘Sur la classification des ensembles de mesure nulle’, Bull. Soc. Math. France 47 (1919), 97–125.Google Scholar | DOI

[3] Devlin, D. C., Some Partition Theorems and Ultrafilters on , Ph.D. thesis, Dartmouth College, Ann Arbor, MI, 1980.Google Scholar

[4] Ehrenfeucht, A. and Mostowski, A., ‘Models of axiomatic theories admitting automorphisms’, Fund. Math. 43 (1956), 50–68.Google Scholar | DOI

[5] Erdős, P. and Rado, R., ‘Combinatorial theorems on classifications of subsets of a given set’, Proc. Lond. Math. Soc. (3) 2 (1952), 417–439.Google Scholar | DOI

[6] Fleissner, W. G., ‘Left separated spaces with point-countable bases’, Trans. Amer. Math. Soc. 294(2) (1986), 665–677.Google Scholar | DOI

[7] Foreman, M., Ideals and Generic Elementary Embeddings, Handbook of Set Theory, Vol. 2 (Springer, Dordrecht, 2010), 885–1147.Google Scholar | DOI

[8] 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.Google Scholar | DOI

[9] Galvin, F., Letter to R. Laver (March 19, 1970).Google Scholar

[10] Gerlits, J. and Szentmiklóssy, Z., ‘A Ramsey-type topological theorem’, Topology Appl. 125(2) (2002), 343–355.Google Scholar | DOI

[11] 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

[12] Jech, T., Set Theory, third millennium edn., revised and expanded, Springer Monogr. Math. (Springer-Verlag, Berlin, 2003).Google Scholar

[13] Jensen, R. and Steel, J., ‘ without the measurable’, J. Symb. Log. 78(3) (2013), 708–734.Google Scholar | DOI

[14] 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

[15] 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.Google Scholar | DOI

[16] 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

[17] Laver, R., ‘On the consistency of Borel’s conjecture’, Acta Math. 137(3-4) (1976), 151–169.Google Scholar | DOI

[18] Magidor, M., ‘How large is the first strongly compact cardinal? or A study on identity crises’, Ann. Math. Logic 10(1) (1976), 33–57.Google Scholar | DOI

[19] Raghavan, D. and Todorcevic, S., ‘Galvin’s problem in higher dimension’, In preparation.Google Scholar

[20] Ramsey, F. P., ‘On a problem of formal logic’, Proc. Lond. Math. Soc. (2) 30(4) (1929), 264–286.Google Scholar

[21] 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

[22] Shelah, S., ‘Was Sierpiński right? IV’, J. Symb. Log. 65(3) (2000), 1031–1054.Google Scholar | DOI

[23] Sierpiński, W., ‘Sur une problème de la théorie des relations’, Ann. Sc. Norm. Super. Pisa (2) 2 (1933), 239–242.Google Scholar

[24] Todorcevic, S., ‘Partition relations for partially ordered sets’, Acta Math. 155(1-2) (1985), 1–25.Google Scholar | DOI

[25] Todorcevic, S., ‘Partitioning pairs of countable ordinals’, Acta Math. 159(3-4) (1987), 261–294.Google Scholar | DOI

[26] Todorcevic, S., ‘A partition property of spaces with point-countable bases’, Unpublished notes (June 1996).Google Scholar

[27] Todorcevic, S., ‘A dichotomy for P-ideals of countable sets’, Fund. Math. 166(3) (2000), 251–267.Google Scholar | DOI

[28] Todorcevic, S., ‘Universally meager sets and principles of generic continuity and selection in Banach spaces’, Adv. Math. 208(1) (2007), 274–298.Google Scholar | DOI

[29] Todorcevic, S., Introduction to Ramsey Spaces, Ann. of Math. Stud., 174 (Princeton University Press, Princeton, NJ, 2010).Google Scholar

[30] Todorcevic, S. and Weiss, W., ‘Partitioning metric spaces’, Unpublished manuscript (September 1995).Google Scholar

[31] Trang, N., ‘ and guessing models’, Israel J. Math. 215(2) (2016), 607–667.Google Scholar | DOI

[32] Woodin, W. H., ‘Supercompact cardinals, sets of reals, and weakly homogeneous trees’, Proc. Natl. Acad. Sci. USA 85(18) (1988), 6587–6591.Google ScholarPubMed | DOI

[33] 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

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