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Erdős’s Work on Infinite Graphs

  • Péter Komjáth
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 25)

Abstract

The theory of infinite graphs was one of Erdös’s favorite topics, and it is no exaggeration to state that the major results and notions were created by him and his collaborators. As one of the few persons equally versed in finite as well as in infinite sets, upon hearing a result on finite graphs, he always eagerly checked if it has a reasonable counterpart for infinite graphs.

Keywords

Bipartite Graph Chromatic Number Triple System Continuum Hypothesis Infinite Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    R. Aharoni: König’s duality theorem for infinite bipartite graphs, Journal of London Mathematical Society, 29 (1984), 1–12.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    R. Aharoni: Menger’s theorem for countable graphs, Journal of Combinatorial Theory (B), 43 (1987), 303–313.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    R. Aharoni, E. Berger: Menger’s theorem for infinite graphs, Inventiones Math., 176 (2009), 1–62.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    J. E. Baumgartner: Generic graph construction, Journal of Symbolic Logic, 49 (1984), 234–240.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    J. E. Baumgartner, A. Hajnal: A remark on partition relations for infinite ordinals with an application to finite combinatorics, in: Logic and combinatorics, Contemporary Mathematics, 65, Amer. Math. Soc., 1987, 157–167.Google Scholar
  6. [6]
    J. Czipszer, P. Erdős, A. Hajnal: Some extremal problems on infinite graphs, Publ. Math. Inst. Hung. Acad.Sci., 7 (1962), 441–456.zbMATHGoogle Scholar
  7. [7]
    N.G. de Bruijn, P. Erdős: A colour problem for infinite graphs and a problem in the theory of relations, Proc. Konink. Nederl. Akad. Wetensch. Amsterdam, 54 (1951), 371–373.Google Scholar
  8. [8]
    W. Deuber: A generalization of Ramsey’s theorem, Infinite and finite sets, (Colloq. Keszthely 1973; dedicated to P. Erdős on his 60th birthday), Vol. I. Colloq. Math. Soc. J. Bolyai, Vol. 10, North Holland, Amsterdam, 1975Google Scholar
  9. [9]
    W. Deuber: Partitionstheoreme für Graphen, Math. Helv., 50 (1975), 311–320.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    A. Dudek, V. Rödl: On the Folkman number f(2; 3; 4), Exp. Math., 17 (2008), 63–67.CrossRefzbMATHGoogle Scholar
  11. [11]
    A. Dudek, V. Rödl: On the Turán properties of infinite graphs, Elect. Journal of Combinatorics, 15 (2008), R47, pp 14.Google Scholar
  12. [12]
    P. Erdős: Some set-theoretical properties of graphs, Revista de la Univ. Nac. de Tucumán, Ser. A. Mat. y Fis. Teór. 3 (1942), 363–367.Google Scholar
  13. [13]
    P. Erdős: Graph theory and probability, Canad. J. Math. 11 (1959), 34–38.CrossRefMathSciNetGoogle Scholar
  14. [14]
    P. Erdős: Problem 8, in: Theory of graphs and its applications, Proceedings of the Symposium held in Smolenice, June 1963, Czechoslovak Acad. Sci. Prague, 1964, p. 159.Google Scholar
  15. [15]
    P. Erdős, F. Galvin, A. Hajnal: On set-systems having large chromatic number and not containing prescribed subsystems, Infinite and finite and sets, (Colloq. Keszthely 1973; dedicated to P. Erdős on his 60th birthday), Vol. I. Colloq. Math. Soc. J. Bolyai, Vol. 10, North Holland, Amsterdam, 1975, 425–513.Google Scholar
  16. [16]
    Erdős Pál, Grünwald Tibor, Weiszfeld Endre: Végtelen gráfok Euler vonalairól, Mat. Fiz. Lapok, 43 (1936), 129–141.zbMATHGoogle Scholar
  17. [17]
    P. Erdős, T. Grünwald, E. Vázsonyi: Über Euler-Linien unendlicher Graphen, J. Math. Physics, 17 (1938), 59–75.Google Scholar
  18. [18]
    P. Erdős, A. Hajnal: On chromatic number of graphs and set-systems, Acta. Math. Hungar., 17 (1966), 61–99.CrossRefGoogle Scholar
  19. [19]
    P. Erdős, A. Hajnal: On decomposition of graphs, Acta. Math. Hungar., 18 (1967), 359–377.CrossRefGoogle Scholar
  20. [20]
    P. Erdős, A. Hajnal: On chromatic number of infinite graphs, in: Theory of graphs, Proc. of the Coll. held at Tihany 1966, Hungary, (ed. P. Erdős and G. Katona), Akadémiai Kiadó, Budapest-Academic Press, New York, 1968, 83–89.Google Scholar
  21. [21]
    P. Erdős, A. Hajnal: Unsolved problems in set theory, in: Axiomatic Set Theory (Proc. Symp. Pure Math. XIII, Part I, Univ. Calif. Los Angeles, Calif. 1967), Amer. Math. Soc., Providence, R.I., 1971, 17–48.Google Scholar
  22. [22]
    P. Erdős, A. Hajnal, L. Pósa: Strong embeddings of graphs into colored graphs, in: Infinite and finite sets, (Colloq. Keszthely 1973; dedicated to P. Erdős on his 60th birthday), Vol. I. Colloq. Math. Soc. J. Bolyai, Vol. 10, North Holland, Amsterdam, 1975, 585–595Google Scholar
  23. [23]
    P. Erdős, A. Hajnal, B. L. Rothschild: On chromatic number of graphs and setsystems, in: Cambridge School in Mathematical Logic (Cambridge, England, 1971), Lecture Notes in Mathematics, Vol. 337, Springer, Berlin, 1973, 531–538.Google Scholar
  24. [24]
    P. Erdős, A. Hajnal, S. Shelah: On some general properties of chromatic numbers, in: Topics in topology (Proc. Colloq. Keszthely, 1972), Colloq. Math. Soc. J. Bolyai, Vol. 8. North Holland, Amsterdam, 1974, 243–255.Google Scholar
  25. [25]
    P. Erdős, A. Hajnal, E. Szemerédi: On almost bipartite large chromatic graphs, Annals of Discrete Math., 12 (1982), 117–123.Google Scholar
  26. [26]
    P. Erdős, S. Kakutani: On non-denumerable graphs, Bull. Amer. Math. Soc. 49 (1943), 457–461.CrossRefMathSciNetGoogle Scholar
  27. [27]
    P. Erdős, R. Rado: A construction of graphs without triangles having pre-assigned order and chromatic number, J. London Math. Soc., 35 (1960), 445–448.CrossRefMathSciNetGoogle Scholar
  28. [28]
    J. Folkman: Graphs with monochromatic complete subgraphs in every edge coloring, SIAM Journ. of Applied Math., 18 (1970), 19–24.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [29]
    M. Foreman: An ℵ1-dense ideal on ℵ2, Israel Journ. Math., 108 (1998), 253–290.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [30]
    M. Foreman, R. Laver: Some downward transfer properties for ℵ2, Advances in Mathematics, 67 (1988), 230–238.CrossRefzbMATHMathSciNetGoogle Scholar
  31. [31]
    F. Galvin: Chromatic numbers of subgraphs, Periodica Math. Hung., 4 (1973), 117–119.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [32]
    A. Hajnal: The chromatic number of the product of two ℵ1-chromatic graphs can be countable, Combinatorica, 5 (1985), 137–140.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [33]
    A. Hajnal: Embedding finite graphs into graphs colored with infinitely many colors, Israel Journal of Math., 73 (1991), 309–319.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [34]
    A. Hajnal: Paul Erdős’ set theory, in: The Mathematics of Paul Erdős, (R. Graham, J. Nešetřil, eds.), Springer, 1997, 352–393.Google Scholar
  35. [35]
    A. Hajnal, P. Komjáth: What must and what need not be contained in a graph of uncountable chromatic number? Combinatorica, 4 (1984), 47–52CrossRefzbMATHMathSciNetGoogle Scholar
  36. [36]
    A. Hajnal, P. Komjáth: Embedding graphs into colored graphs, Trans. Amer. Math. Soc., 307 (1988), 395–409.CrossRefzbMATHMathSciNetGoogle Scholar
  37. [37]
    A. Hajnal, P. Komjáth: Obligatory subsystems of triple systems, Acta Math. Hung., 119 (2008), 1–13.CrossRefzbMATHGoogle Scholar
  38. [38]
    P. Komjáth: Connectivity and chromatic number of infinite graphs, Israel Journal of Mathematics, 56 (1986), 257–266.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [39]
    P. Komjáth: The colouring number, Proc. London Math. Soc., 54 (1987), 1–14.CrossRefzbMATHMathSciNetGoogle Scholar
  40. [40]
    P. Komjáth: Consistency results on infinite graphs, Israel Journal of Mathematics, 61 (1988), 285–294.CrossRefzbMATHMathSciNetGoogle Scholar
  41. [41]
    P. Komjáth: Third note on Hajnal-Máté graphs, Periodica Math. Hung., 24 (1989), 403–406.zbMATHGoogle Scholar
  42. [42]
    P. Komjáth: The chromatic number of some uncountable graphs, Coll. Math. Soc. János Bolyai, 60, Sets, graphs, and numbers, Budapest (Hungary), 1991, 439–444.Google Scholar
  43. [43]
    P. Komjáth: Ramsey-theory and forcing extensions, Proc. Amer. Math. Soc. 121, (1994), 217–219.zbMATHMathSciNetGoogle Scholar
  44. [44]
    P. Komjáth: Two remarks on the coloring number, Journal of Combinatorial Theory, (B), 70 (1997), 301–305.CrossRefzbMATHGoogle Scholar
  45. [45]
    P. Komjáth: Some remarks on obligatory subsytems of uncountably chromatic triple systems, Combinatorica, 21 (2001), 233–238.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [46]
    P. Komjáth: Subgraph chromatic number, DIMACS Series in Discrete Mathematics and Computer Science, 58, 2002, 99–106.Google Scholar
  47. [47]
    P. Komjáth: An uncountably chromatic triple system, Acta Math. Hung., 121 (2008), 79–92.CrossRefzbMATHGoogle Scholar
  48. [51]
    P. Komjáth, S. Shelah: Forcing constructions for uncountably chromatic graphs, Journal of Symbolic Logic, 53 (1988), 696–707.CrossRefzbMATHMathSciNetGoogle Scholar
  49. [52]
    P. Komjáth, S. Shelah: A consistent partition theorem for infinite graphs, Acta Math. Hung., 61 (1993), 115–120.CrossRefzbMATHGoogle Scholar
  50. [53]
    P. Komjáth, S. Shelah: Finite subgraphs of uncountably chromatic graphs, Journal of Graph Theory, 49 (2005), 28–38.CrossRefzbMATHMathSciNetGoogle Scholar
  51. [54]
    D. Kőnig: Theorie der endlichen und unendlichen Graphen, Akademische Verlagsgesellschaft, MBG, Leipzig, 1936.Google Scholar
  52. [56]
    L. Lovász: Combinatorial Problems and Exercises, North-Holland, 1973.Google Scholar
  53. [57]
    J. Nešetřil, V. Rödl: Type theory of partition properties of graphs, in: Recent Advances in Graph Theory, (ed. M. Fiedler), Academia, Prague, 1975, 183–192.Google Scholar
  54. [58]
    J. Nešetřil, V. Rödl: Ramsey properties of graphs with forbidden complete subgraphs, Journ. Comb. Th., (B), 20 (1976), 243–249.CrossRefzbMATHGoogle Scholar
  55. [59]
    J. Nešetřil, V. Rödl: A short proof of the existence of restricted Ramsey graphs by means of a partite construction, Combinatorica, 1 (1981), 199–202.CrossRefzbMATHMathSciNetGoogle Scholar
  56. [60]
    V. Rödl: M.Sc. Thesis, Charles University, Prague, 1973.Google Scholar
  57. [61]
    V. Rödl: On the chromatic number of subgraphs of a given graph, Proc. Amer. Math. Soc., 64 (1977), 370–371.CrossRefzbMATHMathSciNetGoogle Scholar
  58. [62]
    S. Shelah: Infinite abelian groups, Whitehead problem, and some constructions, Israel Journal of Mathematics, 18 (1974), 243–256.CrossRefzbMATHMathSciNetGoogle Scholar
  59. [63]
    S. Shelah: Consistency of positive partition theorems for graphs and models, Set theory and applications (J. Steprāns, S. Watson, eds.), Lecture Notes in Math., 1401, 167–193.Google Scholar
  60. [64]
    S. Shelah: Incompactness for chromatic numbers of graphs, A tribute to P. Erdős (A. Baker, B. Bollobás, A. Hajnal, eds) Camb. Univ. Press, (1990), 361–371.Google Scholar
  61. [65]
    L. Soukup: On chromatic number of product of graphs, Comm. Math. Univ. Carol., 29 (1988), 1–12.zbMATHMathSciNetGoogle Scholar
  62. [66]
    J. Spencer: Three hundred million points suffice, Journ. Comb. Th., (A), 49 (1988), 210–217.CrossRefzbMATHGoogle Scholar
  63. [67]
    C. Thomassen: Cycles in graphs of uncountable chromatic number, Combinatorica, 3 (1983), 133–134.CrossRefzbMATHMathSciNetGoogle Scholar
  64. [68]
    S. Todorcevic: Coloring pairs of countable ordinals, Acta Math., 159 (1987), 261–294.CrossRefzbMATHMathSciNetGoogle Scholar
  65. [69]
    S. Todorcevic: Comparing the continuum with the first two uncountable cardinals. in: Logic and Scientific Methods, (eds. M. L. Dalla Chiara et al.), Kluwer, 1997, 145–155.Google Scholar
  66. [70]
    S. Todorcevic: Combinatorial dichotomies in set theory, Bull. of the Symbolic Logic, 17 (2011), 1–72.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag 2013

Authors and Affiliations

  • Péter Komjáth
    • 1
  1. 1.Institute of MathematicsEötvös UniversityHungary

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