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Ramsey Theory in the Work of Paul Erdős

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The Mathematics of Paul Erdös II

Part of the book series: Algorithms and Combinatorics ((AC,volume 14))

Summary

Ramsey’s theorem was not discovered by P. Erdős. But perhaps one could say that Ramsey theory was created largely by him. This paper will attempt to demonstrate this claim.

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References

  1. P. Erdős and G. Szekeres, A combinatorial problem in geometry, Composito Math. 2 (1935), 464–470.

    Google Scholar 

  2. I. Schur, Über die Kongruens x m + y m = z m(mod p), Jber. Deutch. Math. Verein 25 (1916), 114–117.

    MATH  Google Scholar 

  3. I. Schur, Gesammelte Abhandlungen (eds. A. Brauer, H. Rohrbach), 1973, Springer.

    Google Scholar 

  4. B. L. van der Waerden, Berver’s einer Baudetschen Vermutung, Nieuw. Arch. Wisk. 15 (1927), 212–216.

    Google Scholar 

  5. R. Rado, Studien zur Kombinatorik, Math. Zeitschrift 36 (1933), 242–280.

    Article  MathSciNet  Google Scholar 

  6. F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 48 (1930), 264–286.

    Article  Google Scholar 

  7. P. Erdős, Art of Counting, MIT Press.

    Google Scholar 

  8. R. L. Graham, B. L. Rothschild, and J. Spencer, Ramsey theory, Wiley, 1980, 2nd edition, 1990.

    MATH  Google Scholar 

  9. A. Thomason, An upper bound for some Ramsey numbers, J. Graph Theory 12 (1988), 509–517.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53(1947), 292–294.

    Article  MathSciNet  Google Scholar 

  11. P. Erdős, Graph theory and probability, Canad. J. Math. 11 (1959), 34–38.

    Article  MathSciNet  Google Scholar 

  12. P. Erdős, Graph theory and probabilityII, Canad. J. Math. 13 (1961), 346–352.

    Article  MathSciNet  Google Scholar 

  13. V. Chvätal and J. Komlos, Some combinatorial theorems on monotonicity, Canad. Math. Bull. 14, 2 (1971).

    Article  Google Scholar 

  14. J. Nešetřil and V. Rödl, A probabilistic graph theoretical method, Proc. Amer. Math. Soc. 72 (1978), 417–421.

    MathSciNet  MATH  Google Scholar 

  15. P. Valtr, Convex independent sets and 7-holes in restricted planar point sets, Discrete Comput. Geom. 7 (1992), 135–152.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Nešetřil and P. Valtr, A Ramsey-type result in the plane, Combinatorics, Probability and Computing 3 (1994), 127–135.

    Article  MATH  Google Scholar 

  17. V. Jarnik and M. Kössler, Sur les graphes minima, contenant n points donnés, Čas. Pěst. Mat. 63 (1934), 223–235.

    Google Scholar 

  18. P. Hell and R. L. Graham, On the history of the minimum spanning tree problem, Annals of Hist. Comp. 7 (1985), 43–57.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Hilbert, Über die irreduribilität ganzer rationaler funktionen mit ganssahligen koeffieienten, J. Reine und Angew. Math. 110 (1892), 104–129.

    Article  Google Scholar 

  20. J. H. Spencer, Ramsey’s theorem — a new lower bound, J. Comb. Th. A, 18 (1975), 108–115.

    Article  MATH  Google Scholar 

  21. Blanche Descartes, A three colour problem, Eureka 9 (1947), 21, Eureka 10 (1948)24. (See also the solution to Advanced problem 1526, Amer. Math. Monthly 61 (1954), 352.)

    Google Scholar 

  22. G. A. Dirac, The structure of k-chromatic graphs, Fund. Math. 40 (1953), 42–55.

    MathSciNet  MATH  Google Scholar 

  23. J. B. Kelly and L. M. Kelly, Paths and Circuits in critical graphs, Amer. J. Math. 76 (1954), 786–792.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Mycielski, Sur le coloriage des graphes, Collog. Math. 3 (1955), 161–162.

    MathSciNet  MATH  Google Scholar 

  25. J. Nešetřil, Chromatic graphs without cycles of length ≤ 7, Comment. Math., Univ. Carolina (1966).

    Google Scholar 

  26. A. A. Zykov, On some properties of linear complexes, Math. Sbornik 66, 24 (1949), 163–188.

    MathSciNet  Google Scholar 

  27. P. Erdős and A. Hajnal, On chromatic number of set systems, Acta Math. Acad. Sci. Hungar. 17 (1966), 61–99.

    Article  MathSciNet  Google Scholar 

  28. P. Erdős and A. Hajnal, Some remarks on set theory IX, Mich. Math. J. 11 (1964), 107–112.

    Article  Google Scholar 

  29. P. Erdős and R. Rado, A construction of graphs without triangles having preassigned order and chromatic number, J. London Math. Soc. 35 (1960), 445–448.

    Article  MathSciNet  Google Scholar 

  30. L. Loväsz, On the chromatic number of finite set-systems, Acta Math. Acad. Sci. Hungar. 19 (1968), 59–67.

    Article  MathSciNet  Google Scholar 

  31. J. Nešetřil and V. Rödl, A short proof of the existence of highly chromatic graphs without short cycles, J. Combin. Th. B, 27 (1979), 525–52?.

    Google Scholar 

  32. G. A. Margulis, Explicit constructions of concentrators, Problemy Peredachi Informatsii 9, 4 (1975), 71–80.

    MathSciNet  Google Scholar 

  33. A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan Graphs, Combinatorica 8(3) (1988), 261–277.

    Article  MathSciNet  MATH  Google Scholar 

  34. N. Alon, Eigenvalues, geometric expanders, sorting in sounds and Ramsey theory, Combinatorica 3 (1986), 207–219.

    Article  Google Scholar 

  35. M. Ajtai, J. Komlós and E. Szemerédi, A dense infinite Sidon sequence, European J. Comb. 2 (1981), 1–11.

    MATH  Google Scholar 

  36. P. Erdős, Problems and results in additive number theory, Colloque sur la Theorie des Numbres, Bruxelles (1955), 127–137.

    Google Scholar 

  37. P. Erdős and P. Turán, On a problem of Sidon in additive number theory and on some related problems, J. London Math. Soc. 16 (1941), 212–215.

    Article  MathSciNet  Google Scholar 

  38. S. Sidon, Ein Satz über trigonometrische Polynome und seine Anwendungen in der Theorie der Fourier-Reihen, Math. Ann. 106 (1932), 539.

    Article  MathSciNet  Google Scholar 

  39. J. Nešetřil and V. Rödl, Two proofs in combinatorial number theory, Proc. Amer. Math. Soc. 93, 1 (1985), 185–188.

    MathSciNet  MATH  Google Scholar 

  40. P. Erdős, On sequences of integers no one of which divides the product of two others and on some related problems, Izv. Nanc. Ise. Inset. Mat. Mech. Tomsrk 2 (1938), 74–82.

    Google Scholar 

  41. N. Alon and J. Spencer, Probabilistic methods, Wiley, New York, 1992.

    Google Scholar 

  42. J. H. Kim, The Ramsey number R(3, t)has order of magnitude t 2/log t Random Structures and Algorithms (to appear).

    Google Scholar 

  43. F. R. K. Chung, R. L. Graham and R. M. Wilson, Quasirandom graphs, Combinatorica 9 (1989), 345–362.

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Thomason, Random graphs, strongly regular graphs and pseudorandom graphs, In: Survey in Combinatorics, Cambridge Univ. Press (1987), 173–196.

    Google Scholar 

  45. P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357–368.

    Article  MathSciNet  MATH  Google Scholar 

  46. P. Frankl, A constructive lower bound fo Ramsey numbers, Ars Combinatorica 2 (1977), 297–302.

    MathSciNet  Google Scholar 

  47. P. Erdős and C. A. Rogers, The construction of certain graphs, Canad. J. Math. (1962), 702–707.

    Google Scholar 

  48. D. Preiss and V. Rödl, Note on decomposition of spheres with Hilbert spaces, J. Comb. Th. A 43 (1) (1986), 38–44.

    Article  MATH  Google Scholar 

  49. N. Alon, Explicit Ramsey graphs and orthonormal labellings, Electron. J. Cornbin. 1, R12 (1994), (8pp).

    Google Scholar 

  50. F. R. K. Chung, R. Cleve, and P. Dagum, A note on constructive lower bound for the Ramsey numbers R(3, t), J. Comb. Theory 57 (1993), 150–155.

    Article  MathSciNet  MATH  Google Scholar 

  51. P. Erdős and R. Rado, A combinatorial theorem, J. London Math. Soc. 25 (1950), 249–255.

    Article  MathSciNet  Google Scholar 

  52. P. Erdős and R. Rado, Combinatorial theorems on classifications of subsets of a given set, Proc. London Math. Soc. 3 (1951), 417–439.

    Google Scholar 

  53. P. Erdős and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427–489.

    Article  MathSciNet  Google Scholar 

  54. P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers, Acta Math. Hungar. 16 (1965), 93–196.

    Article  Google Scholar 

  55. R. Rado, Note on canonical partitions, Bull. London Math. Soc. 18 (1986), 123–126. Reprinted: Mathematics of Ramsey Theory (ed. J. Nešetřil and V. Rödl), Springer 1990, pp. 29–32.

    Article  MathSciNet  MATH  Google Scholar 

  56. H. Lefman and V. Rödl, On Erdős-Rado numbers, Combinatorica (1995).

    Google Scholar 

  57. S. Shelah, Finite canonization, 1994 (preprint) (to appear in Comm. Math. Univ. Carolinae).

    Google Scholar 

  58. J. Pelans and V. Rödl, On coverings of infinite dimensional metric spaces. In Topics in Discrete Math., vol. 8 (ed. J. Nešetřil), North Holland (1992), 75–81.

    Google Scholar 

  59. P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, L’ Enseignement Math. 28 (1980), 128 pp.

    Google Scholar 

  60. H. Lefman, A canonical version for partition regular systems of linear equations, J. Comb. Th. A 41 (1986), 95–104.

    Article  Google Scholar 

  61. P. Erdős, J. Nešetřil and V. Rödl, Selectivity of hypergraphs, Colloq. Math. Soc. János Bolyai 37 (1984), 265–284.

    Google Scholar 

  62. R. L. Graham, On edgewise 2-colored graphs with monochromatic triangles and containing no complete hexagon, J. Comb. Th. 4 (1968), 300.

    Article  Google Scholar 

  63. R. Irving, On a bound of Graham and Spencer for a graph-coloring constant, J. Comb. Th. B, 15 (1973), 200–203.

    Article  MathSciNet  MATH  Google Scholar 

  64. M. Erickson, An upper bound for the Folkman number F(3, 3, 5), J. Graph Th. 17 (6), (1993), 679–68.

    Article  MathSciNet  MATH  Google Scholar 

  65. J. Bukor, A note on Folkman number F(3, 3, 5), Math. Slovaka 44 (4), (1994), 479–480.

    MathSciNet  MATH  Google Scholar 

  66. J. Folkman, Graphs with monochromatic complete subgraphs in every edge coloring, SIAM J. Appl. Math. 18 (1970), 19–24.

    Article  MathSciNet  MATH  Google Scholar 

  67. J. Nešetřil and V. Rödl, Type theory of partition properties of graphs, In: Recent Advances in Graph Theory (ed. M. Fiedler), Academia, Prague (1975), 405–412.

    Google Scholar 

  68. P. Erdős, Problems and result on finite and infinite graphs, In: Recent Advances in Graph Theory (ed. M. Fiedler), Academia, Prague (1975), 183–192.

    Google Scholar 

  69. J. Spencer, Three hundred million points suffice, J. Comb. Th. A 49 (1988), 210–217.

    Article  MATH  Google Scholar 

  70. V. Rödl and A. Rucińiski, Threshold functions for Ramsey properties, J. Amer. Math. Soc. (1995), to appear.

    Google Scholar 

  71. J. Nešetřil, Ramsey Theory, In: Handbook of Combinatorics, North Holland (1995), to appear.

    Google Scholar 

  72. P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. H. Spencer, and E. G. Straus, Euclidean Ramsey Theorem, J. Combin. Th. (A) 14 (1973), 341–363.

    Article  Google Scholar 

  73. P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. H. Spencer, and E. G. Straus, Euclidean Ramsey Theorems II, In A. Hajnal, R. Rado and V.Sós, eds., Infinite and Finite Sets I, North Holland, Amsterdam, 1975, pp. 529–557.

    Google Scholar 

  74. P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. H. Spencer, and E. G. Straus, Euclidean Ramsey Theorems III, In A. Hajnal, R. Rado and V. Sós, eds., Infinite and Finite Sets II, North Holland, Amsterdam, 1975, pp. 559–583.

    Google Scholar 

  75. P. Frankl and V. Rödl, A partition property of simplices in Euclidean space, J. Amer. Math. Soc. 3 (1990), 1–7.

    Article  MathSciNet  MATH  Google Scholar 

  76. I. Kříž Permutation groups in Euclidean Ramsey theory, Proc. Amer. Math. Soc. 112 (1991), 899–907.

    MathSciNet  MATH  Google Scholar 

  77. R. L. Graham, Recent trends in Euclidean Ramsey theory, Disc. Math. 136 (1994), 119–127.

    Article  MATH  Google Scholar 

  78. W. Deuber, R. L. Graham, H. J. Prömel and B. Voigt, A canonical partition theorem for equivalence relations on Zt, J. Comb. Th. (A) 34 (1983), 331–339.

    Article  MATH  Google Scholar 

  79. B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1927), 212–216.

    Google Scholar 

  80. B. L. van der Waerden, How the Proof of Baudet’s Conjecture was found, in Studies in Pure Mathematics (ed. L. Mirsky), Academic Press, New York, 1971, pp. 251–260.

    Google Scholar 

  81. A. J. Khinchine, Drei Perlen der Zahlen Theorie, Akademie Verlag, Berlin 1951 (reprinted Verlag Harri Deutsch, Frankfurt 1984).

    Google Scholar 

  82. A. W. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222–229.

    Article  MathSciNet  MATH  Google Scholar 

  83. E. R. Berlekamp, A construction for partitions which avoid long arithmetic progressions, Canad. Math. Bull 11 (1968), 409–414.

    Article  MathSciNet  MATH  Google Scholar 

  84. S. Shelah, Primitive recursive bounds for van der Waerden numbers, J. Amer. Math. Soc. 1 (1988), 683–697.

    Article  MathSciNet  MATH  Google Scholar 

  85. P. Erdős and P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.

    Article  Google Scholar 

  86. R. Salem and D. C. Spencer, On sets of integers which contain no three terms in arithmetic progression, Proc. Nat. Acad. Sci. 28 (1942), 561–563.

    Article  MathSciNet  MATH  Google Scholar 

  87. F. A. Behrend, On sets of integers which contain no three in arithmetic progression, Proc. Nat. Acad. Sci. 23 (1946), 331–332.

    Article  MathSciNet  Google Scholar 

  88. D. Coppersmith and S. Winograd, Matrix multiplication via arithmetic progressions, J. Symb. Comput. 9 (1987), 251–280.

    Article  MathSciNet  Google Scholar 

  89. K. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109.

    Article  MathSciNet  MATH  Google Scholar 

  90. V. Chvátal, V. Rödl, E. Szemerédi, and W. Trotter, The Ramsey number of graph with bounded maximum degree, J. Comb. Th. B, 34 (1983), 239–243.

    Article  MATH  Google Scholar 

  91. J. Nešetřil and V. Rödl, Partition theory and its applications, in Surveys in Combinatorics, Cambridge Univ. Press, 1979, pp. 96–156.

    Google Scholar 

  92. V. Rödl and A. Rucihski, Threshold functions for Ramsey properties, J. Amer. Math. Soc. (1995), to appear.

    Google Scholar 

  93. H. Furstenberg, Ergo die behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math. 31 (1977), 204–256.

    Article  MathSciNet  MATH  Google Scholar 

  94. H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton, 1981.

    MATH  Google Scholar 

  95. H. Furstenberg and Y. Katznelson, A density version of the Hales-Jewett theorem, J. Analyze Math. 57 (1991), 61–85.

    MathSciNet  Google Scholar 

  96. V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden’s and Szemerédi’s theorems, J. Amer. Math. Soc. (1995), to appear.

    Google Scholar 

  97. P. Erdős, J. Nešetřil, and V. Rödl, On Pisier Type Problems and Results (Combinatorial Applications to Number Theory). In: Mathematics of Ramsey Theory (ed. J. Nešetřil and V. Rödl), Springer Verlag (1990), 214–231.

    Google Scholar 

  98. P. Erdős, J. Nešetřil, and V. Rödl, On Colorings and Independent Sets (Pisier Type Theorems) (preprint).

    Google Scholar 

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Authors and Affiliations

  1. AT&T Bell Laboratories, Murray Hill, NJ 07974, USA

    R. L. Graham

  2. Charles University, Praha, The Czech Republic

    J. Nešetřil

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  1. R. L. Graham
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Editors and Affiliations

  1. AT&T Bell Laboratories, 600 Mountain Avenue, 07974, Murray Hill, NJ, USA

    Ronald L. Graham

  2. Department of Applied Mathematics, Charles University, Malostranske nam. 25, 11800, Praha, Czech Republic

    Jaroslav Nešetřil

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Graham, R.L., Nešetřil, J. (1997). Ramsey Theory in the Work of Paul Erdős. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös II. Algorithms and Combinatorics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60406-5_16

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  • DOI: https://doi.org/10.1007/978-3-642-60406-5_16

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