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The Mathematics of Paul Erdös II pp 333–351Cite as

The Random Graph

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

Summary

Erdős and Rényi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.

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References

  1. P. Bankston and W. Ruitenberg (1990), Notions of relative ubiquity for invariant sets of relational structures, J. Symbolic Logic 55, 948–986.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Blass and F. Harary (1979), Properties of almost all graphs and complexes, J. Graph Theory 3, 225–240.

    Article  MathSciNet  MATH  Google Scholar 

  3. B. Bollobás (1985), Random Graphs, Academic Press, London.

    MATH  Google Scholar 

  4. P. J. Cameron (1976), Transitivity of permutation groups on unordered sets, Math. Z. 148, 127–139.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. J. Cameron (1987), On the structure of a random sum-free set, Probab. Thy. Rel Fields 76, 523–531.

    Article  MATH  Google Scholar 

  6. P. J. Cameron (1990), Oligomorphic Permutation Groups, London Math. Soc. Lecture Notes 152, Cambridge Univ. Press, Cambridge.

    Book  MATH  Google Scholar 

  7. P. J. Cameron (1992), The age of a relational structure, pp. 49–67 in Directions in Infinite Graph Theory and Combinatorics(ed. R. Diestel), Topics in Discrete Math 3, North-Holland, Amsterdam.

    Google Scholar 

  8. P. J. Cameron and P. Erdős (1990), On the number of sets of integers with various properties, pp. 61–79 in Number Theory (ed. R. A. Mollin), de Gruyter, Berlin.

    Google Scholar 

  9. P. J. Cameron and K. W. Johnson (1987), An investigation of countable B-groups, Math. Proc. Cambridge Philos. Soc. 102, 223–231.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. J. Cameron and C. Martins (1993), A theorem on reconstructing random graphs, Combinatorics, Probability and Computing 2, 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Cantor (1895), Beiträge zur Begründung der transfiniten Mengenlehre, Math. Ann. 46, 481–512.

    Article  Google Scholar 

  12. G. L. Cherlin (1987), Homogeneous directed graphs, J. Symbolic Logic, 52, 296. (See also Memoirs Amer. Math. Soc. to appear.)

    Google Scholar 

  13. W. Deuber (1975), Partitionstheoreme für Graphen, Math. Helvetici 50, 311–320.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Droste (1985), Structure of partially ordered sets with transitive automorphism groups, Mem. Amer. Math. Soc. 57.

    Google Scholar 

  15. M. El-Zahar and N. W. Sauer (to appear), Ramsey-type properties of relational structures, Discrete Math

    Google Scholar 

  16. E. Engeler (1959), Äquivalenz von n-Tupeln, Z. Math. Logik Gründl. Math. 5, 126–131. [?]

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Erdös, D. J. Kleitman and B. L. Rothschild (1977), Asymptotic enumeration of K n -free graphs, pp. 19–27 in Colloq. Internat. Teorie Combinatorie, Accad. Naz. Lincei, Roma.

    Google Scholar 

  18. P. Erdös and A. Renyi (1963), Asymmetrie graphs, Acta Math. Acad. Sci. Hungar. 14, 295–315

    Article  MathSciNet  Google Scholar 

  19. P.Erdő and J.Spencer(1974), Probabilistic Methods in Combinatorics, Academic Press, New York/Akad. Kiadó Budapest

    Google Scholar 

  20. R. Fagin (1976), Probabilities on finite models, J. Symbolic Logic 41, 50–58.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Fraïssé (1953), Sur certains relations qui généralisent l’ ordre des nombres rationnels, C. R. Acad. Sci. Paris 237, 540–542.

    MathSciNet  MATH  Google Scholar 

  22. R. Fraïssé (1986), Theory of Relations North-Holland, Amsterdam.

    MATH  Google Scholar 

  23. A. Gardiner (1976), Homogeneous graphs, J. Combinatorial Theory(B) 20, 94–102.

    Article  MathSciNet  MATH  Google Scholar 

  24. Y. V. Glebskii, D. I. Kogan, M. I. Liogon’kii and V. A. Talanov (1969), Range and degree of realizability of formulas in the restricted predicate calculus, Kiber-netika 2, 17–28.

    MathSciNet  Google Scholar 

  25. F. Hausdorff (1914), Grundzügen de Mengenlehre, Leipzig.

    Google Scholar 

  26. C. W. Henson (1971), A family of countable homogeneous graphs, Pacific J. Math. 38, 69–83.

    MathSciNet  Google Scholar 

  27. C. W. Henson (1972), Countable homogeneous relational structures and No-categorical theories, J. Symbolic Logic 37, 494–500.

    Article  MathSciNet  MATH  Google Scholar 

  28. W. A. Hodges (1993), Model Theory, Cambridge Univ. Press, Cambridge.

    Book  MATH  Google Scholar 

  29. W. A. Hodges, I. M. Hodkinson, D. Lascar and S. Shelah (1993), The small index property for a;-stable, cj-categorical structures and for the random graph, J. London Math. Soc.(2) 48, 204–218.

    Article  MathSciNet  MATH  Google Scholar 

  30. E. Hrushovski (1992), Extending partial isomorphisms of graphs, Combinatorica 12, 411–416.

    Article  MathSciNet  MATH  Google Scholar 

  31. E. Hrushovski (1993), A new strongly minimal set, Ann. Pure Appl. Logic 62, 147–166.

    Article  MathSciNet  MATH  Google Scholar 

  32. E. V. Huntington (1904), The continuum as a type of order: an exposition of the model theory, Ann. Math. 6, 178–179.

    MathSciNet  Google Scholar 

  33. P. Komjâth and J. Pach (1992), Universal elements and the complexity of certain classes of graphs, pp. 255–270 in Directions in Infinite Graph Theory and Combinatorics(ed. R. Diestel), Topics in Discrete Math. 3, North-Holland, Amsterdam.

    Google Scholar 

  34. A. H. Lachlan (1984), Countable homogeneous tournaments, Trans. Amer. Math. Soc. 284, 431–461.

    Article  MathSciNet  MATH  Google Scholar 

  35. A. H. Lachlan and R. E. Woodrow (1980), Countable ultrahomogeneous undirected graphs, Trans Amer. Math. Soc 262, 51–94.

    Article  MathSciNet  MATH  Google Scholar 

  36. A. H. Mekler (1986), Groups embeddable in the autohomeomorphisms of Q, J. London Math. Soc.(2) 33, 49–58.

    Article  MathSciNet  MATH  Google Scholar 

  37. J. Nešetřil and V. Rödl (1978), The structure of critical Ramsey graphs, Colloq. Internat. C.N.R.S. 260, 307–308.

    Google Scholar 

  38. P. M. Neumann (1985), Automorphisms of the rational world, J. London Math. Soc.(2) 32, 439–448.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. C. Oxtoby (1971), Measure and Category, Springer, Berlin.

    MATH  Google Scholar 

  40. J. Pach (1981), A problem of Ulam on planar graphs, Europ. J. Combinatorics 2, 351–361.

    MathSciNet  Google Scholar 

  41. J. Plotkin (to appear), Who put the back in back-and-forth?

    Google Scholar 

  42. R. Rado (1964), Universal graphs and universal functions, Acta Arith 9, 393–407.

    MathSciNet  Google Scholar 

  43. R. Rado (1967), Universal graphs, in A Seminar in Graph Theory(ed. F. Harary and L. W. Beineke), Holt, Rinehard & Winston, New York.

    Google Scholar 

  44. C. Ryll-Nardzewski (1959), On the categoricity in power N0, Bull. Acad. Polon. Sci. Ser. Math. 7, 545–548.

    MathSciNet  MATH  Google Scholar 

  45. J. H. Schmerl (1979), Countable homogeneous partially ordered sets, Algebra Universalis 9, 317–321.

    Article  MathSciNet  MATH  Google Scholar 

  46. J. J. Seidel (1977), A survey of two-graphs, pp. 481–511 in Colloq. Internat. Teorie Combinatories, Accad. Naz. Lincei, Roma.

    Google Scholar 

  47. S. Shelah and J. Spencer (1988), Zero-one laws for sparse random graphs, J. American. Math. Soc. 1, 97–115.

    Article  MathSciNet  MATH  Google Scholar 

  48. W. Sierpiński (1920), Une propriété topologique des ensembles dénombrables denses en soi, Fund. Math. 1, 11–16.

    Google Scholar 

  49. L. Svenonius (1955), No-categoricity in first-order predicate calculus, Theoria 25, 82–94.

    Article  MathSciNet  Google Scholar 

  50. S. Thomas (1991), Reducts of the random graph, J. Symbolic Logic 56, 176–181.

    Article  MathSciNet  MATH  Google Scholar 

  51. J. K. Truss (1985), The group of the countable universal graph, Math. Proc. Cambridge Philos. Soc. 98, 213–245.

    Article  MathSciNet  MATH  Google Scholar 

  52. J. K. Truss (1989), Infinite permutation groups, I, Products of conjugacy classes, J. Algebra 120, 454–493; II, Subgroups of small index, ibid 120, 494–515.

    Article  MathSciNet  MATH  Google Scholar 

  53. J. K. Truss (1992), Generic automorphisms of homogeneous structures, Proc. London Math. Soc.(3) 65, 121–141.

    Article  MathSciNet  MATH  Google Scholar 

  54. F. O. Wagner (1994), Relational structures and dimensions, Automorphisms of First-Order Structures(ed. R. Kaye and H. D. Macpherson), Oxford University Press, pp. 153–180.

    Google Scholar 

  55. P. Winkler (1993), Random structures and zero-one laws, Finite and Infinite Combinatorics in Sets and Logic(ed. N. W. Sauer, R. E. Woodrow and B. Sands), NATO Advanced Science Institutes Series, Kluwer Academic Publishers, pp. 399–420.

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London, El 4NS, UK

    Peter J. Cameron

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  1. Peter J. Cameron
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Editor information

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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© 1997 Springer-Verlag Berlin Heidelberg

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Cameron, P.J. (1997). The Random Graph. 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_32

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  • DOI: https://doi.org/10.1007/978-3-642-60406-5_32

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64393-4

  • Online ISBN: 978-3-642-60406-5

  • eBook Packages: Springer Book Archive

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