On the combinatorial problems which I would most like to see solved

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References

  1. [1]

    M. Ajtai, J. Komlós andE. Szemerédi, On infinite Sidon sequences,European J. Comb., to appear.

  2. [2]

    K. Appel andW. Haken, Every planar map is four colourable,Illinois J. of Math. 21 (1977) 429–490.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    F. A. Behrend, On sets of integers which contain no three terms in arithmetic progression,Proc. Nat. Acad. Sci. USA 32 (1946) 331–332.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    C. Berge,Graphs and Hypergraphs, North Holland Publ. Comp. Amsterdam—London 1973.

    Google Scholar 

  5. [5]

    B. Bollobás,Extremal Graph Theory, London Math. Soc. Mon. No. 11. 1978.

  6. [6]

    J. A. Bondy, Reflections on the legitimate deck problem,Lecture Notes in Math.,686, Springer 1977, 1–12

    MathSciNet  Article  Google Scholar 

  7. [7]

    J. A. Bondy andP. Erdős, Ramsey numbers for cycles in graphs,J. Comb. Theory 14 (1973) 46–54.

    MATH  Article  Google Scholar 

  8. [8]

    J. A. Bondy andR. L. Hemminger, Graph reconstruction,J. of Graph Theory 1 (1977), 227–268.

    MATH  MathSciNet  Google Scholar 

  9. [9]

    R. C. Bose, E. T. Parker andS. Shrikhande, Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture,Can. J. Math. 12, 189–203.

  10. [10]

    W. G. Brown, On graphs that do not contain a Thomsen graph,Canad. Math. Bull. 9 (1966) 281–285.

    MATH  MathSciNet  Google Scholar 

  11. [11]

    W. G. Brown, P. Erdős andV. T. Sós, Some extremal problems onr-graphs,New Directions in the Theory of Graphs (ed. F. Harary), Academic Press 1973, 53–64.

  12. [12]

    W. G. Brown, P. Erdős andV. T. Sós, On the existence of triangulated spheres in 3-graphs and related questions,Periodica Math. Hung. 3 (1973) 221–228.

    MATH  Article  Google Scholar 

  13. [13]

    R. H. Bruck andH. J. Ryser, The nonexistence of certain projective planes,Canad. J. Math. 1 (1949) 88–93.

    MATH  MathSciNet  Google Scholar 

  14. [14]

    N. G. de Bruijn andP. Erdős, A colour problem for infinite graphs and a problem in theory of relations,Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951) 371–373.

    MATH  Google Scholar 

  15. [15]

    S. A. Burr, Generalized Ramsey theory for graphs — a survey,Graphs and Combinatorics, Lecture Notes in Math. 406, Springer 1974, 52–75.

    Article  MathSciNet  Google Scholar 

  16. [16]

    P. A. Catlin, Subgraphs of graphs I,Discrete Math. 10 (1974) 225–233.

    MATH  Article  MathSciNet  Google Scholar 

  17. [17]

    V. Chvátal, Intersection families of edges in hypergraphs having the hereditary property,Hypergraph Seminar, Lecture Notes in Math. 411, Springer 1972, 61–66.

    Article  Google Scholar 

  18. [18]

    V. Chvátal, R. L. Graham, H. A. Perold andS. H. Whitesides, Combinatorial designs related to the strong perfect graph conjecture,Discrete Math. 26 (1979) 83–92.

    MATH  Article  MathSciNet  Google Scholar 

  19. [19]

    J. Dénes andA. D. Keedwell,Latin Squares and their applications, Academic Press, New York 1974.

    Google Scholar 

  20. [20]

    G. Dirac, In abstrakten Graphen vorhandene 4-Graphen und ihre Unterteilungen,Math. Nachrichten 22 (1960) 61–85.

    MATH  MathSciNet  Google Scholar 

  21. [21]

    P. Erdős, Combinatorial problems in geometry and number theory,to appear. See alsoP. Erdős, On some problems of elementary and combinatorial geometry,Annali di Mat. ser. 4,104 (1975) 99–108.

    Article  Google Scholar 

  22. [22]

    P. Erdős andS. Fajtlowicz, On the conjecture of Hajós,to appear in Combinatorica.

  23. [23]

    P. Erdős, A. W. Goodman andL. Pósa, The representation of a graph by set intersection,Canad. J. Math. 18 (1966) 106–112.

    MathSciNet  Google Scholar 

  24. [24]

    P. Erdős andA. Hajnal, Unsolved and solved problems in set theory,Proc. Symp. Pure Math. XXIII. Univ. Calif. Berkeley, A.M.S. 1971, 17–48.

  25. [25]

    P. Erdős andA. Hajnal, On chromatic number of graphs and set systems,Acta Math. Acad. Sci. Hung. 17 (1966) 61–99.

    Article  Google Scholar 

  26. [26]

    P. Erdős andA. Hajnal, On complete topological subgraphs of certain graphs,Annales Univ. Sci. Budapest 7 (1969) 193–199.

    Google Scholar 

  27. [27]

    P. Erdős, A. Hajnal andE. C. Milner, A problem on well ordered sets,Acta Math. Acad. Sci. Hung. 20 (1969) 323–329.

    Article  Google Scholar 

  28. [28]

    P. Erdős, A. Hajnal andS. Shelah, On some general properties of chromatic numbers,Coll. Math. Soc. J. Bolyai 8,Topics in Topology, North-Holland 1972, 243–255.

    Google Scholar 

  29. [29]

    P. Erdős, A. Hajnal andE. Szemerédi, On almost bipartite large chromatic graphs,to appear in the volume dedicated to the 60th birthday of A. Kotzig.

  30. [30]

    P. Erdős andI. Kaplansky, The asymptotic number of Latin rectangles,Amer. J. Math. 69 (1946) 230–236.

    Article  Google Scholar 

  31. [31]

    P. Erdős andD. Kleitman, Extremal problems among subsets of a set,Discrete Math. 8/3 (1974) 281–294. See alsoG. O. H. Katona, Extremal problems for hypergraphs,Combinatorics (M. Hall and J. H. van Lint, eds.) Math. Centre Amsterdam 1974, 215–244.

    Article  Google Scholar 

  32. [32]

    P. Erdős andL. Lovász, Problems and results on 3-chromatic hypergraphs and some related questions,Coll. Math. J. Bolyai 10 Infinite and Finite Sets, 1973, 609–617.

    Google Scholar 

  33. [33]

    P. Erdős, E. C. Milner andR. Rado, Intersection theorems for systems of sets III.Austral. J. Math. 18 (1974) 22–40.

    Google Scholar 

  34. [34]

    P. Erdős andR. Rado, Intersection theorems for systems of sets,J. London Math. Soc. 35 (1960) 85–90.

    Article  MathSciNet  Google Scholar 

  35. [35]

    P. Erdős andA. Rényi, On the existence of a factor of degree one of a connected random graph,Acta Math. Acad. Sci. Hung. 17 (1966) 359–379.

    Article  Google Scholar 

  36. [36]

    P. Erdős, A. Rényi andV. T. Sós, On a problem of graph theory,Studia Sci. Math. Hung. 1 (1966) 215–235.

    Google Scholar 

  37. [37]

    P. Erdős andM. Simonovits, Some extremal problems in graph theory,Comb. Theory and Appl. (P. Erdős et al. eds.)Coll. Math. Soc. J. Bolyai 4 (1969) 377–390.

    Google Scholar 

  38. [38]

    P. Erdős andM. Simonovits, An extremal graph problem,Acta Math. Acad. Sci. Hung. 22/3–4 (1971) 275–282.

    Google Scholar 

  39. [39]

    P. Erdős andM. Simonovits, On a valence problem in extremal graph theory,Discrete Math. 5 (1973) 323–334.

    Article  MathSciNet  Google Scholar 

  40. [40]

    P. Erdős andM. Simonovits, On the chromatic number of geometric graphs, to appear inArs Combinatoria.

  41. [41]

    P. Erdős andE. Szemerédi, Combinatorial properties of systems of sets,J. of Comb. Theory A 24 (1978) 308–313.

    Article  Google Scholar 

  42. [42]

    P. Erdős andP. Turán, On some sequences of integers,J. of London Math. Soc. 11 (1936) 341–363.

    Google Scholar 

  43. [43]

    P. Erdős andP. Turán, On a problem of Sidon in additive number theory and related problems,J. London Math. Soc. 16 (1941) 212–216.

    Article  MathSciNet  Google Scholar 

  44. [44]

    R. J. Faudree andR. H. Schelp, All Ramsey numbers for cycles in graphs,Discrete Math. 8 (1974) 313–329.

    MATH  Article  MathSciNet  Google Scholar 

  45. [45]

    P. Frankl, On families of finite sets no two of which intersect in a singleton,Bull. Austral. Math. Soc. 17 (1977) 125–134.

    MATH  MathSciNet  Google Scholar 

  46. [46]

    P. Frankl, An intersection problem for finite sets,Acta Math. Acad. Sci. Hung. 30 (1977) 371–373.

    Article  MathSciNet  Google Scholar 

  47. [47]

    P. Frankl, Extremal problems and coverings of the space,European J. of Comb. 1 (1980),to appear.

  48. [48]

    H. Fürstenberg, Ergodic behaviour of diagonal measures and a theorem of Szemerédi on arithmetic progressions I–III,J. Analyse Math. 31 (1977) 204–256.

    MathSciNet  Google Scholar 

  49. [49]

    H. Fürstenberg andY. Katznelson, An ergodic Szemerédi theorem for commuting transformations,J. Analyse Math. 34 (1978) 275–291.

    MathSciNet  Google Scholar 

  50. [50]

    P. Gács andL. Lovász, Khachian’s Algorithm for linear programming,Math. Progr. Studies, to appear

  51. [51]

    T. Gallai, On covering of graphs,Theory of Graphs, Proc. Coll. Tihany, Hungary 1968, (P. Erdős and G. Katona, eds.) 231–236.

  52. [52]

    R. L. Graham, B. Rothschild andJ. Spencer,Ramsey Theory, John Wiley, New York 1980.

    Google Scholar 

  53. [53]

    H. Hadwiger, Ungelöste Probleme No. 40,Elemente der Math. 16 (1961) 103–104.

    MathSciNet  Google Scholar 

  54. [54]

    H. Hadwiger, Über eine Klassifikation der Streckenkomplexe,Vierte Naturforsch. Ges. Zürich 88 (1943) 133–142.

    MathSciNet  Google Scholar 

  55. [55]

    G. Hajós, Über eine Konstruktion nichtn-färbbarer Graphen,Wiss. Zeitschr. Martin Luther Univ. Halle—Wittenberg A 10 (1961) 116–117.

    Google Scholar 

  56. [56]

    H. Hanani, The existence and construction of balanced incomplete block designs,Ann. Math. Stat. 32 (1961) 361–386.

    MathSciNet  MATH  Google Scholar 

  57. [57]

    G. O. H. Katona,unpublished.

  58. [58]

    L. M. Kelly andW. Moser, On the number of ordinary lines determined byn points,Canad. J. Math. 10 (1958) 210–219.

    MATH  MathSciNet  Google Scholar 

  59. [59]

    L. G. Khachiyan, A polynomial algorithm in linear programming (in Russian)Doklady Akademii Nauk SSSR,244, (1979), 1093–1096.

    MATH  MathSciNet  Google Scholar 

  60. [60]

    J. Komlós andE. Szemerédi, Limit distribution for the existence of Hamilton cycles in a random graph,Discrete Math., to appear.

  61. [61]

    T. Kővári, V. T. Sós andP. Turán, On a problem of Zarankiewicz,Coll. Math. 3 (1954) 50–57.

    Google Scholar 

  62. [62]

    D. G. Larman andC. A. Rogers, The realization of distances within sets in Euclidean space,Mathematica 19 (1972) 1–24.

    MathSciNet  MATH  Google Scholar 

  63. [63]

    L. Lovász, Normal hypergraphs and the perfect graph conjecture,Discrete Math. 2 (1972) 253–267.

    MATH  Article  MathSciNet  Google Scholar 

  64. [64]

    L. Lovász,

  65. [65]

    W. Mader, Homomorphieeigenschaften und mittlere Kantendichte von Graphen,Math. Annalen 174 (1967) 265–268.

    MATH  Article  MathSciNet  Google Scholar 

  66. [66]

    U. S. R. Murty, How many magic configurations are there,Amer. Math. Monthly 1978/9, 1001–1002.

  67. [67]

    U. S. R. Murty,unpublished. SeeL. Caccetta andR. Häggkvist, On diameter critical graphs,Discrete Math. 28 (1979) 223–229.

    MathSciNet  Google Scholar 

  68. [68]

    J. Paris andL. Harrington, A Mathematical Incompleteness in Peano Arithmetic,Handbook of Math. Logic, North Holland, Amsterdam 1977, 1133–1142.

    Google Scholar 

  69. [69]

    J. Pelikán, Valency conditions for the existence of certain subgraphs,Theory of Graphs, Proc. Coll. Tihany, Hungary 1968, 251–258.

  70. [70]

    L. Pósa, Hamiltonian cycles in random graphs,Discrete Math. 14 (1976) 359–364.

    MATH  Article  MathSciNet  Google Scholar 

  71. [71]

    V. Rödl,to appear.

  72. [72]

    Vera Rosta, On a Ramsey type problem of J. A. Bondy and P. Erdős I–II,J. Comb. Theory B 15 (1973) 94–105, 105–120.

    MATH  Article  MathSciNet  Google Scholar 

  73. [73]

    K. F. Roth, On certain sets of integers II,J. of London Math. Soc. 29 (1954) 20–26.

    MATH  Article  Google Scholar 

  74. [74]

    I. Z. Ruzsa andE. Szemerédi, Triple systems with no six points carrying three triangles,Coll. Math. Soc. J. Bolyai 18 Combinatorics (A. Hajnal and V. T. Sós, eds.) North-Holland 1978, 939–946.

  75. [75]

    M. Simonovits, Paul Turán*rss influence on graph theory,J. Graph Theory 2 (1977). 102–116.

    MathSciNet  Google Scholar 

  76. [76]

    M. Simonovits, A method for solving extremal problems in graph theory,Theory of Graphs, Proc. Coll. Tihany, Hungary 1966, 279–319.

  77. [77]

    J. Spencer, Intersection theorems for systems of sets,Canad. Math. Bull. 20 (1977) 249–254.

    MATH  MathSciNet  Google Scholar 

  78. [78]

    G. Szekeres, On an extremum problem in the plane,Amer. J. Math. 63 (1941) 208–211.

    MATH  Article  MathSciNet  Google Scholar 

  79. [79]

    E. Szemerédi, On sets of integers containing nok elements in arithmetic progression,Acta Arithm. 27 (1975) 199–245.

    MATH  Google Scholar 

  80. [80]

    E. Szemerédi, A. Gyárfás andZs. Tuza, Induced subtrees in graphs of large chromatic number,Discrete Math. 30 (1980) 235–244.

    MATH  Article  MathSciNet  Google Scholar 

  81. [81]

    P. Turán, On an extremal problem in graph theory (in Hungarian)Mat. Fiz. Lapok 48 (1941) 436–452 and On theory of graphs,Coll. Math. 3 (1954) 19–30.

    MATH  MathSciNet  Google Scholar 

  82. [82]

    B. L. van der Waerden, Beweis einer Baudetschen Vermutung,Nieuw. Arch. Wisk. 15 (1928) 212–216.

    Google Scholar 

  83. [83]

    R. M. Wilson, An existence theory for pairwise balanced designs I–II,J. Comb. Theory A 13 (1972) 220–273.

    MATH  Article  Google Scholar 

  84. [84]

    N. Wormald, A 4-chromatic graph with a special plane drawing,J. Austral. Math. Soc. Ser. A. 28 (1979) 1–8.

    MATH  MathSciNet  Article  Google Scholar 

  85. [85]

    T. Bang, On matrix functioner som med et numerisk lille deficit viser v. d. Waerdens permanenthypotese,Proc. Scandinavian Congress, Turku 1976

  86. [86]

    S. Friedland, A lower bound for the permanent of a doubly stochastic matrix,Ann. Math. 110 (1979), 167–176

    Article  MathSciNet  Google Scholar 

  87. [87]

    B. Grünbaum,Arrangements and Spreads, A. M. S., Providence, R. I. 1972

    Google Scholar 

  88. [88]

    T. S. Motzkin, The lines and planes connecting the points of a finite set,Trans. A. M. S. 70 (1951), 451–464.

    MATH  Article  MathSciNet  Google Scholar 

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Erdős, P. On the combinatorial problems which I would most like to see solved. Combinatorica 1, 25–42 (1981). https://doi.org/10.1007/BF02579174

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AMS subject classification (1980)

  • 05–02
  • 05 C 65, 05 C 35, 05 C 15, 05 B 05, 05 C 55, 05 B 25, 05 B 15, 04 A 20, 10 A 99