Advertisement

Recent Developments in Combinatorial Geometry

  • W. Moser
  • J. Pach
Chapter
Part of the Algorithms and Combinatorics book series (AC, volume 10)

Abstract

Over a span of fifty years Paul Erdős has written many articles with this or a similar title. His countless results, which were obtained by the application of combinatorial and counting (random) methods, and the many deep problems raised and popularized in these papers, generated much research in combinatorics and graph theory. They played an important role in the emergence of a number of new areas in mathematics. One of these is combinatorial geometry, the study of extremal problems about finite arrangements of points, lines, circles, etc.

Keywords

Chromatic Number Unit Distance Geometric Graph Combinatorial Geometry Elementary Cycle 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akiyama, J. and N. Mon (1989): Disjoint simplices and geometric hypergraphs. Combinatorial Mathematics. Proc. Third Internat. Conference (G. Bloom, R. Graham, J. Malkevitch, eds.). Ann. New York Acad. Sci. 555, 1–3Google Scholar
  2. Alon, N. and P. Erdös (1989): Disjoint edges in geometric graphs. Discr. Comput. Geom. 4, 287–290zbMATHCrossRefGoogle Scholar
  3. Alon, N., Z. Fúredi and M. Katchalski (1985): Separating pairs of points by standard boxes. Europ. J. Combinatorics 6, 205–210 [MR 87b: 05011]zbMATHGoogle Scholar
  4. Alon, N. and E. Györi (1986): The number of small semispaces of a finite set of points in the plane. J. Combinat. Th. Ser. A 41, 154–157 [MR 87f: 52014]zbMATHCrossRefGoogle Scholar
  5. Alon, N. and M. Perles (1990): Personal communicationGoogle Scholar
  6. Andreev, E. M (1970a): On convex polyhedra in Lobacevskii spaces. Math. Sbornik USSR, Nov. Ser. 81,445–478Google Scholar
  7. Andreev, E. M (1970b): On convex polyhedra of finite volume in Lobacevskii spaces. Math. Sbornik USSR, Nov. Ser. 83, 256–260Google Scholar
  8. Aronov, B., B. Chazelle, H. Edelsbrunner, L. Guibas, M. Sharir and R. Wenger (1991): Points and triangles in the plane and halving planes in space. Discr. Computat. Geom., to appearGoogle Scholar
  9. Avis, D (1984): The number of farthest neighbour pairs of a finite planar set. Amer. Math. Monthly 91, 417–420. [MR 85j: 52012 Zbl 571.51009]MathSciNetzbMATHCrossRefGoogle Scholar
  10. Avis, D., P. Erdös and J. Pach (1988): Repeated distances in space. Graphs and Combinatorics 4, 207–217. [Zbl 656.05039]MathSciNetzbMATHCrossRefGoogle Scholar
  11. Avis, D., P. Erdös and J. Pach (1991): Distinct distances determined by subsets of a pointset in space. Computat. Geometry, Theory and Applications 1, 1–11zbMATHGoogle Scholar
  12. Bárány, I., Z. Füredi and L. Lovász (1989): On the number of halving planes. Proc. 5th Ann. Sympos. Computat. Geom. 1989: 140–144. To appear also in Combina-torica 10 (1990)Google Scholar
  13. Beck, J (1983): On the lattice property of the plane and some problems of Dirac, Motzkin and Erdös in combinatorial geometry. Combinatorica 3, 281–297. [Zbl 533.52004]MathSciNetzbMATHCrossRefGoogle Scholar
  14. Beck, J. and J. Spencer 1984. Unit distances. J. Combinat. Theory Ser A 37, 231–238. [Zbl 551.05005]MathSciNetzbMATHCrossRefGoogle Scholar
  15. Bellantoni, S., I. Ben-Arroyo Hartman, T. Przytycka and S. Whitesides. 1990. Graphs and boxicity. PreprintGoogle Scholar
  16. Berg, M. T. de and M. J. van Kreveld (1989): Finding squares and rectangles in sets of points. Univ. of Utrecht, Dept. Computer Science Technical Report RUU-CS89–10Google Scholar
  17. Bollobás, B (1978): Extremal Graph Theory. Academic Press, London, New YorkzbMATHGoogle Scholar
  18. Borwein, P. and W. O. J. Moser (1990): A survey of Sylvester’s problem and its generalizations. Aequationes Math. 40, 111–135MathSciNetzbMATHCrossRefGoogle Scholar
  19. Brass, P (1992): The maximum number of second smallest distances in finite planar graphs. Discr. Computat. Geom. 7, 371–379MathSciNetzbMATHCrossRefGoogle Scholar
  20. Buckley, F. and F. Harary (1988): On the Euclidean dimension of a wheel. Graphs and Combinatorics 4, 23–30. [Zbl 642.05018]MathSciNetzbMATHCrossRefGoogle Scholar
  21. Capoyleas, V. and J. Pach (1991): A Turin-type theorem on chords of a convex polygon. J. Combinat. Theory, ser. B, to appearGoogle Scholar
  22. Chazelle, B., E. Edelsbrunner, L. Guibas, R. Pollack, R. Seidel, M. Sharir and J. Snoeyink (1992): Counting and cutting cycles of lines and rods in space. Computational Geom., Theory and Appl. 1, 305–323zbMATHGoogle Scholar
  23. Chung, F. R. K (1984): The number of different distances determined by n points in the plane. J. Combinat. Theory Ser A 36, 342–354. [Zbl 536.05003]zbMATHCrossRefGoogle Scholar
  24. Chung, Fan (1989): Sphere-and-point incidence relations in high dimensions with applications to unit distances and furthest neighbour pairs. Discr. Computat. Geometry 4, 183–190. [Zbl 662.52005]zbMATHCrossRefGoogle Scholar
  25. Chung, F. R. K., E. Szemerédi and W. T. Trotter (1992): The number of different distances determined by a set of points in the Euclidean plane. Discr. Computat. Geom. 7, 1–11zbMATHCrossRefGoogle Scholar
  26. Clarkson, K. L., H. Edelsbrunner, L. J. Guibas, M. Sharir and E. Welzl (1988): Combinatorial complexity bounds for arrangements of curves and surfaces. IEEE Proc. 29th Symposium on Foundations of Computer Science: 568–579. Also in Discr. Computat. Geom. 5, (1990) 99–160Google Scholar
  27. Clarkson, K. L. and P. W. Shor (1989): Applications of random sampling in computational geometry, II. Discr. Computat. Geom. 4, 387–421MathSciNetzbMATHCrossRefGoogle Scholar
  28. Colin de Verdière, Y (1989): Empilements de cercles: Convergence d’une méthode de point fixe. Forum Math. 1, 395–402MathSciNetzbMATHCrossRefGoogle Scholar
  29. Colin de Verdière, Y (1990): Un principe variationnel pour les empilements de cercles. Prépublication de l’Institut Fourier, Laboratoire de Math. No. 147Google Scholar
  30. Conway, J. H., H. T. Croft, P. Erdös and M. J. T. Guy (1979): On the distribution of values of angles by coplanar points. J. London Math. Soc.(2) 19, 137–143. [MR 80h: 51021]MathSciNetzbMATHCrossRefGoogle Scholar
  31. Cozzens, M. B. and F. S. Roberts (1983): Computing the boxicity of a graph by covering its complement by cointerval graphs. Discr. Appl. Math. 6, 217–228. [MR 85d: 05142]MathSciNetzbMATHCrossRefGoogle Scholar
  32. Croft, H. T (1967): Some geometrical thoughts II. Math. Gazette 51, 125–129zbMATHCrossRefGoogle Scholar
  33. Crowe, D. W. and T. A. Mckee (1968): Sylvester’s problem on collinear points. Math. Mag. 41, 30–34. [MR 38 #3761]MathSciNetzbMATHCrossRefGoogle Scholar
  34. Csima, J. and E. T. Sawyer (1993): A short proof that there exist 6n/13 ordinary points. Discr. Computat. Geom., to appearGoogle Scholar
  35. Dirac, G. A (1951): Collinearity properties of sets of points. Quart. J. Math. 2, 221–227. [MR 13 p. 270]MathSciNetzbMATHCrossRefGoogle Scholar
  36. Edelsbrunner, H (1987): Algorithms in Combinatorial Geometry. Springer, Heidelberg. [MR 89a: 68205]zbMATHGoogle Scholar
  37. Edelsbrunner, H. and P. Hajnal (1991): A lower bound on the number of unit distances between the points of a convex polygon. J. Combinat. Theory, Ser. A 56, 312–316MathSciNetzbMATHCrossRefGoogle Scholar
  38. Edelsbrunner, H. and M. Sharir (1990): A hyperplane incidence problem with applications to counting distances. In: Algorithms (Proc. Internat. Symp. SIGAL ‘80, Tokyo. T. Asano et al, eds.) Lecture Notes in Comp. Sci. 450, 419–428. SpringerGoogle Scholar
  39. Edelsbrunner, H and S. Skiena (1989): On the number of furthest neighbour pairs in a point set. Amer. Math. Monthly 96, 614–618MathSciNetzbMATHCrossRefGoogle Scholar
  40. Edelsbrunner, H. and G. Stöckl (1986): The number of extreme pairs of finite point-sets in Euclidean spaces. J. Combinat. Theory Ser A 43, 344–349. [MR 87k: 52025. Zbl 611.51002]zbMATHCrossRefGoogle Scholar
  41. Edelsbrunner, H. and A. Welzl (1985): On the line-separations of a finite set in the plane. J. Combinat. Theory Ser A 38, 15–29. [MR 86m: 52012. Zbl 616.52003]MathSciNetzbMATHCrossRefGoogle Scholar
  42. Erdös, P (1944): Solution of Problem 4065. Amer. Math. Monthly 51, 169–171MathSciNetCrossRefGoogle Scholar
  43. Erdös, P (1946): On sets of distances of n points. Amer. Math. Monthly 53, 248–250. [MR 7 — 471] (This paper is also in: Erdös, P. 1973. The Art of Counting. MIT Press. Cambridge, Mass.) [MR 7 — 471]Google Scholar
  44. Erdös P (1960): On sets of distances of n points in Euclidean space. Publ. Math. Inst. Hungar. Acad. Sci. 5, 165–169. [MR 25 #4420]zbMATHGoogle Scholar
  45. Erdös, P (1965): On extremal problems for graphs and generalized graphs. Israel J. Math. 2, 183–190Google Scholar
  46. Erdös P (1967): On some applications of graph theory to geometry. Canad. J. Math. 19, 968–971. [MR 36 #2520]zbMATHGoogle Scholar
  47. Erdös, P (1987): Some combinatorial and metric problems in geometry. Intuitive Ge-ometry: 167–177. Colloq. Math. Soc. J. Bolyai 48, North-Holland. [Zbl 625.52008]Google Scholar
  48. Erdös, P., Z. Füredi, J. Pach and I. Z. Ruzsa (1993): The grid revisited. Discrete Math. 110, to appearGoogle Scholar
  49. Erdös, P., F. Harary and W. T. Tutte (1965): On the dimension of a graph, Mathematika 12, 118–122. [MR 32 #5537]MathSciNetzbMATHCrossRefGoogle Scholar
  50. Erdös, P., D. Hickerson and J. Pach (1989): A problem of Leo Moser about repeated distances on the sphere. Amer. Math. Monthly 96, 569–575MathSciNetzbMATHCrossRefGoogle Scholar
  51. Erdös, P., L. Lovász, A. Simmons and E. G. Straus (1973): Dissection graphs of planar point sets. A Survey of Combinatorial Theory (J. N. Shrivastava et al, eds.): 139–149. North Holland. [MR 51 #241]Google Scholar
  52. Erdös, P., L. Lovász and K. Vesztergombi (1987): The chromatic number of the graph of large distances. Colloquia Math. Soc. J. Bolyai 52, Combinatorics, Eger (Hungary): 547–551, North-HollandGoogle Scholar
  53. Erdös, P., L. Lovász and K. Vesztergombi (1989): On the graph of large distances. Discr. Computat. Geom. 4, 541–549zbMATHCrossRefGoogle Scholar
  54. Erdös, P., E. Makai, J. Pach and J. Spencer (1991): Gaps in difference sets and the graph of nearly equal distances. In: Applied Geometry and Discrete Mathematics, The Victor Klee Festschrift (P. Gritzmann, B. Sturmfels, eds.), DIMACS Series, Vol. 4, AMS-ACM, 265–273Google Scholar
  55. Erdös, P. and L. Moser (1959): Problem 11. Canad Math. Bull. 2 43Google Scholar
  56. Erdös, P. and J. Pach (1990): Variations on the theme of repeated distances. Cornbinatorica 10, 261–269zbMATHCrossRefGoogle Scholar
  57. Erdös, P. and G. Purdy (1971): Some extremal problems in geometry. J. Combinat. Theory A 10, 246–252. [MR 43 #1045]zbMATHCrossRefGoogle Scholar
  58. Erdös, P. and G. Purdy (1975): Some extremal problems in geometry III. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975): 291–308. Congressus Numerantium XIV, Utilitas Math., Winnipeg, Man. [MR 52 #13650]Google Scholar
  59. Erdös, P. and G. Purdy (1976): Some extremal problems in geometry. IV. Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1976): 307–322. Congressus Numerantium XVII, Utilitas Math., Winnipeg, Man. [MR 55 #10292]Google Scholar
  60. Erdös, P. and M. Simonovits (1980): On the chromatic number of geometric graphs. Ars Combinatoria 9, 229–246. [Zbl 466.05031]MathSciNetzbMATHGoogle Scholar
  61. Erdös, P. and A. H. Stone (1946): On the structure of linear graphs. Bull. Amer. Math. Soc. 52, 1087–1091. [MR 8 — 333]MathSciNetzbMATHCrossRefGoogle Scholar
  62. Frankl, P. and H. Maehara (1986): Embedding the n-cube in lower dimensions. Europ. J. Combinat. 7, 221–225. [Zbl 627.05038]MathSciNetzbMATHGoogle Scholar
  63. Frankl, P. and H. Maehara (1988): On the contact dimension of graphs. Discr. Computat. Geom. 3, 89–96. [Zbl 625.05048]MathSciNetzbMATHCrossRefGoogle Scholar
  64. Frankl, P. and H. Maehara (1988a): The Johnson-Lindenstrauss lemma and the sphericity of some graphs. J. Combinat. Theory Ser. B 44, 355–362. [MR 89e: 05078]MathSciNetzbMATHCrossRefGoogle Scholar
  65. Frankl, P. and R. Wilson (1981): Intersection theorems with geometric consequences. Combinatorica 1, 357–368. [Zbl 498.05046]MathSciNetzbMATHCrossRefGoogle Scholar
  66. Fraysseix, H. de, J. Pach and R. Pollack (1990): Drawing a planar graph on a grid. Combinatorica 10, 41–51MathSciNetzbMATHCrossRefGoogle Scholar
  67. Fúredi, Z. (1990): The maximum number of unit distances in a convex n—gon. J. Combinat. Theory ser. A 55, 316–320CrossRefGoogle Scholar
  68. Gärtner, B. (1989): in a letter from Emo Welzl to Janos Pach.Google Scholar
  69. Gamble, A. B. and M. Katchalski (1986): Separating pairs of points by convex sets. ManuscriptGoogle Scholar
  70. Goodman, J. E. and R. Pollack (1984): On the number of k-subsets of a set of n points in the plane. J. Combinat. Theory Ser. A. 36, 101–104. [MR 85d: 05015]MathSciNetzbMATHCrossRefGoogle Scholar
  71. Gritzmann, P., B. Mohar, J. Pach and R. Pollack (1989): Problem E 3341. Amer. Math. Monthly 96, 642MathSciNetCrossRefGoogle Scholar
  72. Grúnbaum, B. (1963): On Steinitz’s theorem about noninscribable polyhedra. Proc. Ned. Akad. Wetenschap Ser. A 66, 452–455Google Scholar
  73. Griinbaum, B. (1967): Convex Polytopes. Interscience, New YorkGoogle Scholar
  74. Gutman, I. (1977): A definition of dimensionality and distance for graphs. Geometric Representation of Relational Data: 713–723. (J. C. Lingoes, ed.) Mathesis Press, MichiganGoogle Scholar
  75. Gyárfás, A. (1985): On the chromatic number of multiple intervals graphs. Discr. Math. 55, 161–166. [MR 86k: 05052] (Corregendum, ibid 55, 193–204. [MR 87k: 05079])zbMATHCrossRefGoogle Scholar
  76. Hadwiger, H. (1961): Ungelöste Probleme No. 40. Elemente der Math. 16, 103–104MathSciNetGoogle Scholar
  77. Hansen, S. (1981): Contributions to the Sylvester-Gallai Theory. Ph.D.Thesis. CopenhagenGoogle Scholar
  78. Harborth, H. (1974): Solution to problem 664a. Elem. Math 29, 14–15MathSciNetGoogle Scholar
  79. Havel, T. H. (1982): The combinatorial distance geometry approach to the calculation of molecular conformation. Ph. D. Thesis, Group in Biophysics, University of California, BerkeleyGoogle Scholar
  80. Hopf, H. and E. Pannwitz (1934): Aufgabe No. 167. Jber. Deutsch. Math. Verein. 43, 114Google Scholar
  81. Józsa, S. and E. Szemerédi (1975): On the number of unit distances on the plane. Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdös on his 60th birthday), Vol. II: 939–950. Colloq. Math. Soc. Janos Bolyai, Vol. 10, North-Holland, Amsterdam. [MR 52 #10624]Google Scholar
  82. Kashin, B. S. and S. V. Konyagin (1983): On systems of vectors in a Hilbert space. Proc. Steklov Inst. Math. (AMS Translation) Issue 3: 67–70Google Scholar
  83. Kelly, L. M. (1986): A resolution of the Sylvester-Gallai problem of J.-P. Serre. Discr. Computat. Geom. 1, 101–104zbMATHCrossRefGoogle Scholar
  84. Kelly, L. M. and W. Moser (1958): On the number of ordinary lines determined by n points. Canad. J. Math. 10, 210–219 [MR20 #3494]MathSciNetzbMATHCrossRefGoogle Scholar
  85. Koebe, K. (1936): Kontaktprobleme der konformen Abbildung. Berichte über die Verhandlungen der Sächsischen Akad. d. Wissenschaften, Math.-Physikalische Klasse 88, 141–164Google Scholar
  86. Konyagin, S. V. (1981): Systems of vectors in Euclidean space and an extremal problem for polynomials. Mathematicheskie Zametki 29/1, 63–74MathSciNetzbMATHGoogle Scholar
  87. Köviri, T., V. T. Sos and P. Turin (1954): On a problem of K. Zarankiewicz. Colloq. Math. 3, 50–57. [MR 16 — 456]Google Scholar
  88. Kostochka, A. V. (1988): On upper bounds on the chromatic number off graphs (Russian). Trudy Inst. Math. Akad. Nauk SSSR, Sibirskoie Otdel. 10, 204–226MathSciNetGoogle Scholar
  89. Kupitz, Y. (1984): On pairs of disjoint segments in convex position in the plane. Annals Discr. Math. 20, 203–208MathSciNetGoogle Scholar
  90. Kupitz, Y. (1990a): On the maximal number of appearances of the minimal distance among n points in the plane. ManuscriptGoogle Scholar
  91. Kupitz, Y. (1990b): k-supporting hyperplanes of a finite set in d-space. ManuscriptGoogle Scholar
  92. Kupitz, Y. (1990c): k-bisectors of a planar set. Combinatorica, to appearGoogle Scholar
  93. Kupitz, Y. (1990d): Separation of a finite set in d-space by spanned hyperplanes. ManuscriptGoogle Scholar
  94. Kupitz, Y. and M. Perles (1990): The maximal cardinality of a k-thin set in the plane. ManuscriptGoogle Scholar
  95. Larman, D. G. and C. A. Rogers (1972): The realization of distances within sets in Euclidean space. Mathematika 19, 1–24. [MR 47 #7601]MathSciNetzbMATHCrossRefGoogle Scholar
  96. Lovász, L. (1971): On the number of halving lines. Ann. Univ. Sci. Budapest Eötvös, Sect. Math. 14, 107–108Google Scholar
  97. Lovász, L. (1979a): On the Shannon capacity of a graph. IEEE Trans. Infom. Theory 25, 1–7. [MR 81g: 05095]zbMATHCrossRefGoogle Scholar
  98. Lovász, L. (1979b): Combinatorial Problems and Exercises. Akadémiai Kiadó — North-HollandzbMATHGoogle Scholar
  99. Maehara, H. (1984a): Space graphs and sphericity. Discr. Appl. Math. 7, 55–64. [MR 85i: 05092]MathSciNetzbMATHGoogle Scholar
  100. Maehara, H. (1984b): On the sphericity for the join of many graphs, Discr. Math. 49, 311–313. [MR 85i: 05093]MathSciNetzbMATHCrossRefGoogle Scholar
  101. Maehara, H. (1985): Contact patterns of equal nonoverlapping spheres. Graphs and Combinatorics 1, 271–282. [Zbl 581.05022]MathSciNetzbMATHCrossRefGoogle Scholar
  102. Maehara, H. (1988): On the Euclidean dimension of a complete multipartite graph. Discr. Math. 72, 285–289MathSciNetzbMATHCrossRefGoogle Scholar
  103. Maehara, H. (1989): Note on induced subgraphs of the unit distance graph E n. Discr. Computat. Geometry 4, 15–18. [MR 89k: 05035]MathSciNetzbMATHCrossRefGoogle Scholar
  104. Moser, L. (1952): On different distances determined by n points. Amer. Math. Monthly 59, 85–91. [MR 13 — 768]zbMATHCrossRefGoogle Scholar
  105. Moser, L. and W. Moser (1961): Solution P. 10. Canad. Math. Bull. 4, 187–189Google Scholar
  106. O’Donnel, P. and M. Perles (1990): Personal communicationGoogle Scholar
  107. Pach, J. (1980): Decomposition of multiple packing and covering. Diskrete Geometrie, 2 Kolloq. Math. Inst. Univ. Salzburg: 169–178Google Scholar
  108. Pach, J, R. Pollack and E. Welzl (1990): Weaving patterns of lines and line segments in space. Proc. SIGAL Conf. on Algorithms, Tokyo 1990. Lecture Notes in Comp. Sci. 450, 439–446. SpringerGoogle Scholar
  109. Pach, J. and M. Sharir (1992): Repeated angles in the plane and related problems. J. Combinat. Theory ser. A. 59, 12–22MathSciNetzbMATHCrossRefGoogle Scholar
  110. Pach, J., W. Steiger and E. Szemerédi (1989): An upper bound on the number of planar k-sets. Proc. 30th Ann. IEEE Symp. Found. Comput. Sci.: 72–79. Also in: Discrete and Computat. Geometry 7 (1992) 109–123Google Scholar
  111. Pach, J. and J. Töröcsik (1992): Some geometric applications of Dilworth’s theorem. Discr. Computat. Geom., submittedGoogle Scholar
  112. Paturi, R. and J. Simon (1984): Probalistic communication complexity. 25th Annual Symp. on Foundations of Comp. Sc.: 118–126Google Scholar
  113. Perles, M. (1990): Personal communicationGoogle Scholar
  114. Reiterman, J., V. Rödl and E. Sinajovi (1989a): Embeddings of graphs in Euclidean spaces. Discr. Computat. Geom. 4, 349–364zbMATHCrossRefGoogle Scholar
  115. Reiterman, J., V. Rödl and E. Sinajová (1989b): Geometrical embeddings of graphs. Discr. Math. 74, 291–319zbMATHCrossRefGoogle Scholar
  116. Roberts, F. S. (1969a): Indifference graphs. Proof Techniques in Graph Theory: 301–310 (F. Hararay, ed.). Academic Press, New York. [MR 40 #5488]Google Scholar
  117. Roberts, F. S. (1969b): On the boxicity and cubicity of a graph. in Recent Progress in Combinatorics (W. T. Tutte, ed.): 301–310. Academic Press, New York. [MR 40 #5489]Google Scholar
  118. Rosenfeld, M. (1991): Almost orthogonal lines in Ed. In: Applied Geometry and Discrete Mathematics, The Victor Klee Festschrift (P. Gritzmann and B. Sturmfels, eds.), DIMACS Series, Vol. 4, AMS-ACM, 489–492Google Scholar
  119. Schneierman, E. R. and D. B. West (1983): The interval number of a planar graph: three intervals suffice. J. Combinat. Theory Ser. B 35, 224–239. [MR 85e: 05101]CrossRefGoogle Scholar
  120. Schulte, E. (1987): Analogues of Steinitz’s theorem about non-inscribable polytopes. In: Intuitive Geometry (K. Böröczky, G. Fejes Tóth, eds.) Colloq. Math. Soc. J. Bolyai 48, 503–516Google Scholar
  121. Sharir, M. (1991): On joins of lines in 3-space. Submitted to CombinatoricaGoogle Scholar
  122. Spencer, J., E. Szemerédi and W. Trotter Jr. (1984): Unit distances in the Euclidean plane. Graph Theory and Combinatorics, Proc. Conf. hon. P. Erdös, Cambridge 1983: 293–303. Academic Press, London. [Zbl 561.52008. MR 86m: 52015]Google Scholar
  123. Steinitz, E. (1928): Uber isoperimetrische Probleme bei konvexen Polyedern. J. Reine Angew. Math. 159, 133–143zbMATHCrossRefGoogle Scholar
  124. Sylvester, J. J. 1893. Mathematical Question 11851. Educational Times 46, 156Google Scholar
  125. Thurston, W. (1985): The Geometry and Topology of Three-Manifolds. Princeton Notes. Chapter 13Google Scholar
  126. Trotter, W. T. (1979): A characterization of Roberts’ inequality for boxicity, Discr. Math. 28, 303–313. [MR 81a: 05118]MathSciNetzbMATHGoogle Scholar
  127. Trotter, W. T. and F. Harary (1979): On double and multiple interval graphs, J. Graph. Theory 3, 205–211. [MR 81c: 05055]MathSciNetzbMATHCrossRefGoogle Scholar
  128. Turán, P. (1954): On the theory of graphs. Colloq. Math. 3, 19–30.[MR 15–976]zbMATHGoogle Scholar
  129. Vesztergombi, K. (1985): On the distribution of distances in finite sets in the plane. Discr. Math. 57, 129–145. [Zbl 568.52014]MathSciNetzbMATHCrossRefGoogle Scholar
  130. Vesztergombi, K. (1987): On large distances in planar sets. Discr. Math. 67, 191–198. [Zbl 627.52009]MathSciNetzbMATHGoogle Scholar
  131. Woodall, D. R. (1973): Distances realized by sets covering the plane. J. Combinat. Theory Ser. A 14, 187–200. [MR 46 #9868]MathSciNetzbMATHCrossRefGoogle Scholar
  132. Wormald, N. (1979): A 2—chromatic graph with a special plane plane drawing. J. Austral. Math. Soc. A 28, 1–8. [MR 80k: 05060]MathSciNetzbMATHCrossRefGoogle Scholar
  133. Živaljevič, R. T. and S. T. Vrečica (1991): The colored Tverberg problem and complexes of injective functions. manuscriptGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • W. Moser
  • J. Pach

There are no affiliations available

Personalised recommendations