Skip to main content
SpringerLink
Log in

Ramsey Theory Before Ramsey, Prehistory and Early History: An Essay in 13 Parts

  • Chapter
  • First Online:
Book cover Ramsey Theory

Part of the book series: Progress in Mathematics ((PM,volume 285))

Abstract

How does a new theory emerge? It usually manifests itself in the older and established areas of mathematics. Gradually a critical mass of results appears, prompting a realization that what we have is a new identifiable field of mathematical thought, with its own set of problems and methods. As a fetus in a womb, the new theory eventually does not fit in the existing classification of mathematical thought. That is when the child is born. Ramsey theory has not been an exception.

Much of this material is contained in the author’s monograph [Soi], however, this text contains new facts and observations that were not known to the author in 2008 when [Soi] was published.Also, the emphasis here is quite different from [Soi].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    [Sch].

  2. 2.

    Since Van der Waerden was Dutch, I strictly adhere to the Dutch rules in determining where to use “van” and where “Van.” In fact, in Dutch “van” is used only if preceded by the given name(s) or initials.

  3. 3.

    I defer to the celebrated Dutch mathematician N. G. de Bruijn, whose characterization I am quoting here.

  4. 4.

    [Bra2].

  5. 5.

    Quoted from [GRS2].

  6. 6.

    That is, no three points lie on a line.

  7. 7.

    The Mathematical Gazette, 1976.

  8. 8.

    Doklady Akademii Nauk USSR published only papers by full and corresponding members of the Academy. A nonmember’s paper had to be recommended for publication by a full member of the Academy.

  9. 9.

    Theorem 17 also follows from Graham and Rothschild’s results published in 1971 [GR1].

  10. 10.

    As I learned in March 2009, the original exposition was even harder, nearly impenetrable. Ron Graham and Endre Szemerédi spent long hours looking over pages of the proof scattered around in Ron’s house, while simplifying the proof.

  11. 11.

    This Russian publication does not appear in any of Paul Erdős’s bibliographies.

  12. 12.

    We were working on our joint project, a book of Paul’s open problems:Problems of pgom Erdős, which I hope to finish by 2009–2010.

  13. 13.

    This theorem requires the axiom of choice or the equivalent.

References

  1. Arnautov, V.I., Nediskretnaya Topologizuemost Schetnykh Kolets, (Russian), Dokl. Akad. Nauk SSSR 191 (1970), 747–750. English Translation: Nondiscrete topologizability of countable rings, Soviet Math. Dokl. 11 (1970), 423–426.

    Google Scholar 

  2. Brauer, A., Über Sequenzen von Potenzresten, Sitzunberichte de Preussischen Akademie der Wissenschaften, Physicalisch-Mathematische Klasse, Berlin, 1928, 9–16.

    Google Scholar 

  3. Brauer, A., Combinatorial methods in the distribution ofkth power residues, Combinatorial Mathematics and Its Applications; Proceedings of the Conference, University of North Carolina, 14–37, 1969.

    Google Scholar 

  4. Brauer, A., Gedenkrede aufIssai Schur, in Issai Schur. Gesammelte Abhandlungen, vol. 1, Springer, Berlin, 1973, v–xiv.

    Google Scholar 

  5. Bruijn, N.G. de, Letter to P. Erdős, August 18, 1947.

    Google Scholar 

  6. Bruijn, N.G. de and Erdős, P., A colour problem for infinite graphs and a problem in the theory of relations, Indagationes Math. 13 (1951), 369–373.

    Google Scholar 

  7. Burkill, H. and Mirsky, L., Monotonicity, J. Math. Anal. Appl. 41 (1973), 391–410.

    Article  MathSciNet  MATH  Google Scholar 

  8. Chung, F. and Graham, R.L., Forced convex n-gons in the plane, Discrete Comput. Geom. 19 (1998), 3677–371.

    Article  MathSciNet  Google Scholar 

  9. Dickson, L.E., On the Last Theorem of Fermat, Quart. J. Pure Appl. Math. 40 (1908), 27–45.

    MATH  Google Scholar 

  10. Erdős, P., Letter to N. G. de Bruijn, August 4, 1947.

    Google Scholar 

  11. Erdős, P., Some unsolved problems, Michigan Math. J. 4 (1957), 291–300.

    Article  MathSciNet  Google Scholar 

  12. Erdős, P., Some unsolved problems, Magyar tudomanyos akademia mathematikai kutato intezetenek közlemenyel 6 ser. A(1–2) (1961), 221–254.

    Google Scholar 

  13. Erdős, P., Nekotorye Nereshennye Problemy (Some unsolved problems), Matematika 7(4) (1963), 109–143. (A Russian translation of [E61.22].)

    Google Scholar 

  14. Erdős, P., On the application of combinatorial analysis to number theory, geometry and analysis, Actes du Congrès International des Mathematiciens (Nice, 1970), Tome 3, pp. 201–210, Gauthier-Villars, Paris, 1971.

    Google Scholar 

  15. Erdős, P., Problems and results on combinatorial number theory, A survey of Combinatorial Theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colorado, 1971), 117–138, North-Holland, Amsterdam, 1973.

    Google Scholar 

  16. Erdős, P., Problems and results in combinatorial analysis, Colloquio Internacionale sulle Teorie Combinatorie (Rome, 1973), Tomo II, Atti dei Convegni Lincei, No. 17, pp. 3–17, Accad. Naz. Lincei, Rome, 1976.

    Google Scholar 

  17. Erdős, P., On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25–42.

    Article  MathSciNet  Google Scholar 

  18. Erdős, P., Some problems and results in number theory, Number Theory and Combinatorics, Japan 1984 (Tokyo, Okayama and Kyoto, 1984), pp. 65–87, World Scientific, Singapore, 1985.

    Google Scholar 

  19. Erdős, P., Obituary of my friend and coauthor Tibor Gallai, Geombinatorics II(1) (1992), 5–6; corrections: II(2), 1992, 37.

    Google Scholar 

  20. Erdős, P. and Graham, R., Old and New Problems and Results in Combinatorial Number Theory, L’Enseignement Mathématique, Université de Genève, 1980.

    Google Scholar 

  21. Erdős, P. and Szekeres, G., A combinatorial problem in geometry, Compositio Math., 2 (1935), 463–470.

    MathSciNet  Google Scholar 

  22. Erdős, P. and Turán, P., On Some Sequences of Integers, J. Lond. Math. Soc. 11 (1936), 261–264.

    Article  Google Scholar 

  23. Garsia, A., Manuscript, Fall 1958, 10 pp.

    Google Scholar 

  24. Graham, R.L., Recent developments in Ramsey theory. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 1555–1567, PWN, Warsaw, 1984.

    Google Scholar 

  25. Graham, R.L. and Nesetril, J., Ramsey theory and Paul Erdős, in Paul Erdős and his Mathematics II, ed. by G. Halász, et al., Springer, Budapest, 2002, 339–365.

    Google Scholar 

  26. Graham, R.L. and Rothschild, B.L., Ramsey’s theorem for n-parameter sets, Trans. Am. Math. Soc. 159 (1971), 257–292.

    MathSciNet  MATH  Google Scholar 

  27. Graham, R.L., Rothschild, B.L., and Spencer, J.H., Ramsey Theory, 1st Edition, John Wiley & Sons, New York, 1980.

    MATH  Google Scholar 

  28. Graham, R.L., Rothschild, B.L., and Spencer, J.H., Ramsey Theory, 2nd Edition, John Wiley & Sons, New York, 1990.

    MATH  Google Scholar 

  29. [GG] Greenwood, R.E. and Gleason, A.M., Combinatorial relations and chromatic graphs, Can. J. Math. 7 (1955), 1–7.

    Google Scholar 

  30. Hilbert, D., Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 110 (1892), 104–129.

    Google Scholar 

  31. Hindman, N., Finite Sums from Sequences within Cells of a Partition of N, J. Combin. Theory Ser. A 17 (1974), 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  32. Khinchin, A.Y., Tri Zhemchuzhiny Teorii Chisel, Gos. Izd-vo tekhn.-teoret, lit-ry, Moskva, 1947 (Russian).

    Google Scholar 

  33. Khinchin, A.Y., Tri Zhemchuzhiny Teorii Chisel, Gos. Izd-vo tekhn.-teoret, lit-ry, 2nd ed., Moskva, 1948 (Russian).

    Google Scholar 

  34. Chintschin, A.J., Drei Pearlen der Zahlentheorie, Akademie Verlag, Berlin, 1951.

    Google Scholar 

  35. Khinchin, A.Y., Three Pearls of Number Theory, Graylock, Rochester, New York, 1952.

    MATH  Google Scholar 

  36. Mirsky, L., The combinatorics of arbitrary partitions, Bull. Inst. Math. 11 (1975), 6–9.

    MathSciNet  Google Scholar 

  37. Rado, R., Studien zur Kombinatorik, Math. Zeitschrift 36 (1933), 424–480.

    Article  MathSciNet  MATH  Google Scholar 

  38. Rado, R., Note on Combinatorial Analysis, Proc. Lond. Math. Soc. 48(2) (1943), 122–160.

    MathSciNet  MATH  Google Scholar 

  39. Ramsey, F.P., On a problem of formal logic, Proc. Lond. Math. Soc. Ser 2 30(4) (1930), 338–384.

    Google Scholar 

  40. Roth, K.F., On certain sets of integers, J. Lond. Math. Soc. 28 (1953), 104–109.

    Article  MATH  Google Scholar 

  41. Sanders, J.H., A Generalization of Schur’s Theorem, Ph.D. Dissertation, Yale University, 1968.

    Google Scholar 

  42. Schur, I., Über die Kongruenzx m+y m≡z m(mod.p), Jahresbericht der Deutschen Mathematiker-Vereinigung 25 (1916), 114–117.

    Google Scholar 

  43. Seidenberg, A., A simple proof of Erdős and Szekeres, J. London Math. Soc. 34 (1959), 352.

    Google Scholar 

  44. Shelah, S. and Soifer, A., Axiom of choice and chromatic number of the plane, J.Combin. Theory Ser. A 103 (2003), 387–391.

    Article  MathSciNet  MATH  Google Scholar 

  45. Soifer, A. and Shelah, S., Axiom of choice and chromatic number: An example on the plane, J. Combin. Theory Ser. A 2(105) (2004), 359–364.

    Article  MathSciNet  Google Scholar 

  46. Skolem, T., Ein Kombinatorische Satz mit Anwendung auf ein Logisches Entscheidungsproblem, Fundamenta Math. 20 (1933), 254–261.

    MATH  Google Scholar 

  47. Soifer, A., The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators, Springer, New York, 2009.

    MATH  Google Scholar 

  48. Szemerédi, E., On sets of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sci. Hungar. 20 (1969), 89–104.

    Article  MathSciNet  MATH  Google Scholar 

  49. [Sz2] Szemerédi, E., On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica XXVII (1975), 199–245.

    Google Scholar 

  50. Tóth, G. and Valtr, P., The Erdős-Szekeres theorem, upper bounds and generalizations, in Discrete and Computational Geometry – Papers from the MSRI Special Program, ed. by J.E. Goodman et al., MSRI Publications 52, Cambridge University Press, Cambridge, 2005, 557–568.

    Google Scholar 

  51. Waerden, B.L. van der, Beweis einer audetschen Vermutung, Nieuw Archief voor Wiskunde 15 (1927), 212–216.

    Google Scholar 

  52. Waerden, B.L. van der, How the proof of Baudet’s conjecture was found, in Studies in Pure Mathematics, ed. by L. Mirsky, Academic Press, London, 1971, 251–260.

    Google Scholar 

  53. Witt, E., Ein kombinatorischer Satz der Elementargeometrie, Mathematische Nachrichten 6 (1952), 261–262.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Colorado at Colorado Springs, Coloredo Springs, CO, 80918, USA

    Alexander Soifer

Authors
  1. Alexander Soifer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Soifer .

Editor information

Editors and Affiliations

  1. at Colorado Springs, College of Letters, Arts, and Sciences, University of Colorado, Austin Bluffs Parkway 1420, Colorado Springs, 80918, Colorado, USA

    Alexander Soifer

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Soifer, A. (2011). Ramsey Theory Before Ramsey, Prehistory and Early History: An Essay in 13 Parts. In: Soifer, A. (eds) Ramsey Theory. Progress in Mathematics, vol 285. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8092-3_1

Download citation

Publish with us

Policies and ethics

Search

Navigation