Abstract
In this chapter, the following theorem—which can be considered as the nucleus of Ramsey Theory—will be discussed in great detail.
Theorem 4.1 (Ramsey’s Theorem). For any number n ∈ ω, for any positive number r ∈ ω, for any S ∈ [ω]ω, and for any colouring π: [S]n → r, there is always an H ∈ [S]ω such that H is homogeneous for π, i.e., the set [H]n is monochromatic.
Musicians in the past, as well as the best of the moderns, believed that a counterpoint or other musical composition should begin on a perfect consonance, that is, a unison, fifth, octave, or compound of one of these.
Gioseffo Zarlino
Le Istitutioni Harmoniche, 1558
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References
Spiros A. Argyros and Stevo Todorčević, Ramsey Methods in Analysis, [Advanced Courses in Mathematics – CRM Barcelona], Birkhäuser, Basel, 2005.
Taras O. Banakh and Igor V. Protasov, Symmetry and colorings: some results and open problems, Izdatel’stvo Gomel’skogo Universiteta. Voprosy Algebry, vol. 17 (2001), 5–16.
Taras O. Banakh, Oleg V. Verbitski, and Yaroslav B. Vorobets, A Ramsey treatment of symmetry, The Electronic Journal of Combinatorics, vol. 7 #R52 (2000), 25pp.
Jörg Brendle, Lorenz Halbeisen, and Benedikt Löwe, Silver measurability and its relation to other regularity properties, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 138 (2005), 135–149.
Antoine Brunel and Louis Sucheston, B-convex Banach spaces, Mathematical Systems Theory, vol. 7 (1973), 294–299.
Dennis Devlin, Some partition theorems and ultrafilters onω, Ph.D. thesis (1979), Dartmouth College, Hanover (USA).
Paul Erdős, Problems and results in chromatic number theory, in Proof Techniques in Graph Theory (F. Harary, ed.), Academic Press, New York, 1969, pp. 47–55.
Paul Erdős and Richard Rado, A combinatorial theorem, Journal of the London Mathematical Society, vol. 25 (1950), 249–255.
Paul Erdős and George Szekerés, A combinatorial problem in geometry, Compositio Mathematica, vol. 2 (1935), 463–470.
Paul Erdős and András Hajnal, Research problem 2-5, Journal of Combinatorial Theory, vol. 2 (1967), 105.
W. Timothy Gowers, A new dichotomy for Banach spaces, Geometric and Functional Analysis, vol. 6 (1996), 1083–1093.
——, An infinite Ramsey theorem and some Banach-space dichotomies, Annals of Mathematics (2), vol. 156 (2002), 797–833.
——, Ramsey methods in Banach spaces, in Handbook of the geometry of Banach spaces, Vol. 2 (William B. Johnson and Joram Lindenstrauss, eds.), North-Holland, Amsterdam, 2003, pp. 1071–1097.
Ronald L. Graham, On edgewise 2-colored graphs with monochromatic triangles and containing no complete hexagon, Journal of Combinatorial Theory, vol. 4 (1968), 300.
——, Some of my favorite problems in Ramsey theory, in Combinatorial Number Theory (Bruce Landman, Melvyn B. Nathanson, Jaroslav Nešetřil, Richard J. Nowakowski, and Carl Pomerance, eds.), Proceedings of the ‘Integers Conference 2005’ in Celebration of the 70th Birthday of Ronald Graham. Carrollton, Georgia, USA, October 27–30, 2005, Walter de Gruyter, Berlin, 2007, pp. 229–236.
Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer, Ramsey Theory, 2nd ed., J. Wiley & Sons, New York, 1980.
Lorenz Halbeisen, Fans and bundles in the graph of pairwise sums and products, The Electronic Journal of Combinatorics, 11(1) (2004) #R6, 11pp.
——, Making doughnuts of Cohen reals, Mathematical Logic Quarterly, vol. 49 (2003), 173–178.
Lorenz Halbeisen and Norbert Hungerbühler, On generalized Carmichael numbers, Hardy-Ramanujan Journal, vol. 22 (1999), 8–22.
Lorenz Halbeisen and Edward W. Odell, On asymptotic models in Banach spaces, Israel Journal of Mathematics, vol. 139 (2004), 253–291.
Neil Hindman, Finite sums from sequences within cells of a partition of N, Journal of Combinatorial Theory (A), vol. 17 (1974), 1–11.
Neil Hindman and Dona Strauss, Algebra in the Stone-Čech Compactification: Theory and Applications, [De Gruyter Expostitions in Mathematics], Walter de Gruyter, New York, 1998.
Thomas Jech, The Axiom of Choice, Studies in Logic and the Foundations of Mathematics 75, North-Holland, Amsterdam, 1973.
Péter Komjáth, A coloring result for the plane, Journal of Applied Analysis, vol. 5 (1999), 113–117.
Mordechai Lewin, A new proof of a theorem of Erdős and Szekerés, The Mathematical Gazette, vol. 60 (1976), 136–138 (correction ibid., p. 298).
Keith R. Milliken, Ramsey’s theorem with sums or unions, Journal of Combinatorial Theory (A), vol. 18 (1975), 276–290.
Walter D. Morris and Valeriu Soltan, The Erdős-Szekeres problem on points in convex position – a survey, Bulletin of the American Mathematical Society, vol. 37 (2000), 437–458.
Edward W. Odell, Applications of Ramsey theorems to Banach space theory, in Notes in Banach Spaces (H.E. Lacey, ed.), University Press, Austin and London, Austin TX, 1980, pp. 379–404.
——, On subspaces, asymptotic structure, and distortion of Banach spaces; connections with logic, in Analysis and Logic (Catherine Finet and Christian Michaux, eds.), [London Mathematical Society Lecture Note Series 262], Cambridge University Press, Cambridge, 2002, pp. 189–267.
Jeff B. Paris, Combinatorial statements independent of arithmetic, in Mathematics of Ramsey Theory (J. Nešetřil and V. Rödl, eds.), Springer-Verlag, Berlin, 1990, pp. 232–245.
Jeff B. Paris and Leo Harrington, A mathematical incompleteness in Peano arithmetic, in Handbook of Mathematical Logic (J. Barwise, ed.), North-Holland, Amsterdam, 1977, pp. 1133–1142.
Richard Rado, Direct decomposition of partitions, Journal of the London Mathematical Society, vol. 29 (1954), 71–83.
Stanisław P. Radziszowski, Small Ramsey numbers, The Electronic Journal of Combinatorics, Dynamic Surveys 1, (August 1, 2006), 60pp.
Frank P. Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society, Ser. II, vol. 30 (1930), 264–286.
——, The Foundations of Mathematics and other Logical Essays, edited by R. B. Braithwaite, Kegan Paul, Trench, Trubner & Co., London, 1931.
Issai Schur, Über die Kongruenzx m + y m = z m (modp), Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 25 (1916), 114–117.
Neil J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, available online at: http://www.research.att.com/~njas/sequences/.
Alexander Soifer, Issai Schur: Ramsey theory before Ramsey, Geombinatorics, vol. 5 (1995), 6–23.
Alan D. Taylor, A canonical partition relation for finite subsets of ω, Journal of Combinatorial Theory (A), vol. 21 (1976), 137–146.
Stevo Todorčević, Introduction to Ramsey Spaces, [Annals of Mathematics Studies 174], Princeton University Press, Princeton (New Jersey), 2010.
Vojkan Vuksanović, A proof of a partition theorem for \([\mathbb{Q}]^{n},\) Proceedings of the American Mathematical Society, vol. 130 (2002), 2857–2864.
Arthur Wieferich, Zum letzten Fermatschen Theorem, Journal für die Reine und Angewandte Mathematik, vol. 136 (1909), 293–302.
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Halbeisen, L.J. (2017). Overture: Ramsey’s Theorem. In: Combinatorial Set Theory. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-60231-8_4
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