Papers
Topics
Authors
Recent
Assistant
AI Research Assistant
Well-researched responses based on relevant abstracts and paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses.
Gemini 2.5 Flash
Gemini 2.5 Flash 165 tok/s
Gemini 2.5 Pro 50 tok/s Pro
GPT-5 Medium 41 tok/s Pro
GPT-5 High 33 tok/s Pro
GPT-4o 124 tok/s Pro
Kimi K2 193 tok/s Pro
GPT OSS 120B 443 tok/s Pro
Claude Sonnet 4.5 36 tok/s Pro
2000 character limit reached

A Ramsey theorem for the reals (2405.18431v1)

Published 28 May 2024 in math.LO, math.CO, and math.GN

Abstract: We prove that for every colouring of pairs of reals with finitely-many colours, there is a set homeomorphic to the rationals which takes no more than two colours. This was conjectured by Galvin in 1970, and a colouring of Sierpi{\'n}ski from 1933 witnesses that the number of colours cannot be reduced to one. Previously in 1985 Shelah had shown that a stronger statement is consistent with a forcing construction assuming the existence of large cardinals. Then in 2018 Raghavan and Todor\v{c}evi\'c had proved it assuming the existence of large cardinals. We prove it in $ZFC$. In fact Raghavan and Todor\v{c}evi\'c proved, assuming more large cardinals, a similar result for a large class of topological spaces. We prove this also, again in $ZFC$.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (41)
  1. Baumgartner, J. E. Partition relations for countable topological spaces. J. Combin. Theory Ser. A 43, 2 (1986), 178–195.
  2. Devlin, D. C. Some Partition Theorems And Ultrafilters On Omega. ProQuest LLC, Ann Arbor, MI, 1980. Thesis (Ph.D.)–Dartmouth College.
  3. Dobrinen, N. Ramsey theory of homogeneous structures: current trends and open problems. In ICM—International Congress of Mathematicians. Vol. III. Sections 1–4. EMS Press, Berlin, [2023] ©2023, pp. 1462–1486.
  4. When Ramsey Theory fails settle for more colors (big Ramsey degrees!). ACM SIGACT News 51, 4 (2020), 30–46.
  5. Eisworth, T. Galvin’s conjecture and weakly precipitous ideals, 2023.
  6. Unsolved problems in set theory. In Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) (1971), Proc. Sympos. Pure Math., XIII, Part I, Amer. Math. Soc., Providence, RI, pp. 17–48.
  7. Unsolved and solved problems in set theory. In Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) (1974), Proc. Sympos. Pure Math., Vol. XXV, Published for the Association for Symbolic Logic by the American Mathematical Society, Providence, RI, pp. 269–287.
  8. Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. 16 (1965), 93–196.
  9. Combinatorial theorems on classifications of subsets of a given set. Proc. London Math. Soc. (3) 2 (1952), 417–439.
  10. On some problems involving inaccessible cardinals. In Essays on the foundations of mathematics. Magnes Press, The Hebrew University, Jerusalem, 1961, pp. 50–82.
  11. Fleissner, W. G. Left separated spaces with point-countable bases. Trans. Amer. Math. Soc. 294, 2 (1986), 665–677.
  12. Galvin, F. Letter to Richard Laver, March 19, 1970.
  13. Inequalities for cardinal powers. Ann. of Math. (2) 101 (1975), 491–498.
  14. Some counterexamples in the partition calculus. J. Combinatorial Theory Ser. A 15 (1973), 167–174.
  15. Discrete subspaces of topological spaces. II. Indag. Math. 31 (1969), 18–30. Nederl. Akad. Wetensch. Proc. Ser. A 72.
  16. Kanamori, A. The higher infinite, second ed. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. Large cardinals in set theory from their beginnings.
  17. Larson, P. B. The stationary tower, vol. 32 of University Lecture Series. American Mathematical Society, Providence, RI, 2004. Notes on a course by W. Hugh Woodin.
  18. Munkres, J. R. Topology. Prentice Hall, Inc., Upper Saddle River, NJ, 2000. Second edition of [ MR0464128].
  19. Proof of a conjecture of Galvin. Forum Math. Pi 8 (2020), e15, 23.
  20. Galvin’s problem in higher dimensions. Proc. Amer. Math. Soc. 151, 7 (2023), 3103–3110.
  21. Ramsey, F. P. On a Problem of Formal Logic. Proc. London Math. Soc. (2) 30, 4 (1929), 264–286.
  22. Rinot, A. Dushnik-Miller for singular cardinals (part 2). https://blog.assafrinot.com/?p=628. Accessed: 2024-05-14.
  23. Rinot, A. Chain conditions of products, and weakly compact cardinals. Bull. Symb. Log. 20, 3 (2014), 293–314.
  24. Shelah, S. A note on cardinal exponentiation. J. Symbolic Logic 45, 1 (1980), 56–66.
  25. Shelah, S. Was Sierpiński right? I. Israel J. Math. 62, 3 (1988), 355–380.
  26. Shelah, S. Strong partition relations below the power set: consistency; was Sierpiński right? II. In Sets, graphs and numbers (Budapest, 1991), vol. 60 of Colloq. Math. Soc. János Bolyai. North-Holland, Amsterdam, 1992, pp. 637–668. arXiv: math/9201244.
  27. Shelah, S. Cardinal arithmetic, vol. 29 of Oxford Logic Guides. The Clarendon Press, Oxford University Press, New York, 1994.
  28. Shelah, S. Applications of PCF theory. J. Symbolic Logic 65, 4 (2000), 1624–1674. arXiv: math/9804155.
  29. Shelah, S. Was Sierpiński right? IV. J. Symbolic Logic 65, 3 (2000), 1031–1054. arXiv: math/9712282.
  30. Shelah, S. The Erdős-Rado arrow for singular cardinals. Canad. Math. Bull. 52, 1 (2009), 127–131. arXiv: math/0605385.
  31. Sierpiński, W. Sur un problème de la théorie des relations. Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) 2, 3 (1933), 285–287.
  32. Sierpiński, W. Sur une propriété topologique des ensembles dénombrables denses en soi. Fundamenta Mathematicae 1, 1 (1920), 11–16.
  33. Todorcevic, S. Universally meager sets and principles of generic continuity and selection in Banach spaces. Adv. Math. 208, 1 (2007), 274–298.
  34. Todorcevic, S. Walks on ordinals and their characteristics, vol. 263 of Progress in Mathematics. Birkhäuser Verlag, Basel, 2007.
  35. Todorcevic, S. Introduction to Ramsey spaces, vol. 174 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2010.
  36. Todorcevic, S. Basis problems and expansions, III. Fields Institute Seminar (March 17, 2023).
  37. Todorčević, S. Partitioning pairs of countable ordinals. Acta Math. 159, 3-4 (1987), 261–294.
  38. Todorčević, S. Some partitions of three-dimensional combinatorial cubes. J. Combin. Theory Ser. A 68, 2 (1994), 410–437.
  39. Partitioning metric spaces, September 1995.
  40. Weiss, W. Partitioning topological spaces. In Mathematics of Ramsey theory, vol. 5 of Algorithms Combin. Springer, Berlin, 1990, pp. 154–171.
  41. Woodin, W. H. The axiom of determinacy, forcing axioms, and the nonstationary ideal, revised ed., vol. 1 of De Gruyter Series in Logic and its Applications. Walter de Gruyter GmbH & Co. KG, Berlin, 2010.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Dice Question Streamline Icon: https://streamlinehq.com

Open Problems

  1. Galvin’s Conjecture on two-color bound for homeomorphic copies of Q in the reals
Lightbulb Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Authors (1)

  1. Tanmay Inamdar
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets

This paper has been mentioned in 2 tweets and received 15 likes.

Upgrade to Pro to view all of the tweets about this paper:

Start a free 7-day Pro trial