aequationes mathematicae

, Volume 40, Issue 1, pp 111–135

# A survey of Sylvester's problem and its generalizations

• P. Borwein
• W. O. J. Moser
Survey Paper

## Summary

LetP be a finite set of three or more noncollinear points in the plane. A line which contains two or more points ofP is called aconnecting line (determined byP), and we call a connecting lineordinary if it contains precisely two points ofP. Almost a century ago, Sylvester posed the disarmingly simple question:Must every set P determine at least one ordinary line? No solution was offered at that time and the problem seemed to have been forgotten. Forty years later it was independently rediscovered by Erdös, and solved by Gallai. In 1943 Erdös proposed the problem in the American Mathematical Monthly, still unaware that it had been asked fifty years earlier, and the following year Gallai's solution appeared in print. Since then there has appeared a substantial literature on the problem and its generalizations.

In this survey we review, in the first two sections, Sylvester's problem and its generalization to higher dimension. Then we gather results about the connecting lines, that is, the lines containing two or more of the points. Following this we look at the generalization to finite collections of sets of points. Finally, the points will be colored and the search will be for monochromatic connecting lines.

### AMS (1980) subject classification

Primary 51A20 Secondary 52A37

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### References

1. Balomenos, R. H., Bonnice, W. E. andSilverman, R. J. 1966.Extensions of Sylvester's theorem. Canad. Math. Bull.9, 1–14. MR:33, # 6477.Google Scholar
2. Basterfield, J. G. andKelly, L. M. 1968.A characterization of sets of n points which determine n hyperplanes. Proc. Cambridge Philos. Soc.64, 585–588. MR:38, # 2040.Google Scholar
3. Baston, V. J. andBostock, F. A. 1978.A Gallai-type problem. J. Combin. Theory Ser. A24, 122–125. MR:57, # 150.
4. Beck, J. 1983.On a “lattice property” of the plane and some problems in combinatorial geometry. Combinatorica3, 281–297. Zbl 533.52004.Google Scholar
5. Bonnice, W. andEdelstein, M. 1967.Flats associated with finite sets in P d. Niew Arch. Wisk.15, 11–14. MR:35, # 7189.Google Scholar
6. Bonnice, W. andKelly, L. M. 1971.On the number of ordinary planes. J. Combin. Theory11, 45–53. MR:45, # 1784.
7. Boros, E., Füredi Z. andKelly, L. M. On representing Sylvester-Gallai designs. Discrete Comput. Geom.4, 345–348.Google Scholar
8. Borwein, J. 1979.Monochrome lines in the plane. Math. Mag.52, 41–45.Google Scholar
10. Borwein, P. 1982.On monochrome lines and hyperplanes. J. Combin. Theory Ser. A33, 76–81.
11. Borwein, P. 1983a.On Sylvester's problem and Haar spaces. Pacific J. Math.109, 275–278.Google Scholar
12. Borwein, P. 1983b.The Desmic conjecture. J. Combin. Theory Ser. A35, 1–9.
13. Borwein, P. 1984.Sylvester's problem and Motzkin's theorem for countable and compact sets. Proc. Amer. Math. Soc.90, 580–584.Google Scholar
14. Borwein, P. andEdelstein, M. 1983.A conjecture related to Sylvester's problem. Amer. Math. Monthly90, 389–390.
15. Brakke, K. A. 1972.Some new values of Sylvester's function for n noncollinear points. J. Undergrad. Math.4, 11–14.Google Scholar
16. Bruijn, N. G. de andErdös, P. 1948.On a combinatorial problem. Neder. Akad. Wetensch. Proc.51, 1277–1279 (=Indagationes Math10, 421–423). MR:10, p. 424.Google Scholar
17. Burckhardt, J. J. 1940.Über konvexe Körper mit Mittlepunkt. Vierteljahrsschr. Naturforsch: Ges. Zürich85, 149–154.Google Scholar
18. Burr, S. A., Grünbaum, B. andSloane.N. J. A. 1974.The Orchard Problem. Geom. Dedicata2, 397–424. MR:49, # 2428.
19. Chakerian, G. D. 1970.Sylvester's problem on collinear points and a relative. Amer. Math. Monthly77, 164–167. MR:41, # 3305.Google Scholar
20. Coxeter, H. S. M. 1984.A problem of collinear points. Amer. Math. Monthly55, 26–28. MR:9, p. 458.Google Scholar
21. Coxeter, H. S. M. 1955.The Real Projective Plane (2nd ed.). Cambridge University Press, Cambridge. MR:10, p. 729.Google Scholar
22. Coxeter, H. S. M. 1968.Twelve Geometric Essays. Southern Illinois University Press, Carbondale, IL. MR:46, # 9843.Google Scholar
23. Coxeter, H. S. M. 1969.Introduction to Geometry (2nd ed.). Wiley, New York. MR:49, # 11369.Google Scholar
24. Coxeter, H. S. M. 1973.Regular Polytopes (3rd ed.). Dover, New York. MR:51, # 6554.Google Scholar
25. Croft, H. T. 1967.Incidence incidents. Eureka (Cambridge)30, 22–26.Google Scholar
26. Crowe, D. W. andMcKee, T. A. 1968.Sylvester's problem on collinear points. Math. Mag.41, 30–34. MR:38, # 3761.Google Scholar
27. Dirac, G. A. 1951.Collinearity properties of sets of points. Quart. J. Math.2, 221–227. MR:13, p. 270.Google Scholar
28. Dirac, G. A. 1959.Review of Kelly and Moser (1958). MR:20, # 3494.Google Scholar
29. Edelstein, M. 1957.A further generalization of a problem of Sylvester. Riveon Lamatematika11, 50–55. Hebrew. English summary. MR:21, # 2211.Google Scholar
30. Edelstein, M. 1969.Hyperplanes and lines associated with families of compact sets in locally convex spaces. Math. Scand.25, 25–30. MR:41, # 983.Google Scholar
31. Edelstein, M., Herzog, F. andKelly, L. M. 1963.A further theorem of the Sylvester type. Proc. Amer. Math. Soc.14, 359–363. MR:26, # 4333.Google Scholar
32. Edelstein, M. andKelly, L. M. 1966.Bisecants of finite collections of sets in linear spaces. Canad. J. Math.18, 375–380. MR:32, # 6185.Google Scholar
33. Edmonds, J., Lovász, andMandel, A. 1980.Solution to problem in vol. 1, p. 250. Math. Intelligencer2, 106–107.Google Scholar
34. Erdös, P. 1943.Problem 4065. Amer. Math. Monthly50, 65.Google Scholar
35. Erdös, P. 1944.Solution of Problem 4065. Amer. Math. Monthly51, 169–171.Google Scholar
36. Erdös, P. 1972.On a problem of Grünbaum. Canad. Math. Bull.15, 23–25. MR:47, # 5709.Google Scholar
37. Erdös, P., 1973.Problems and results in combinatorial analysis. (Technion Preprint Series No. MT-182). Technion, Haifa.Google Scholar
38. Erdös, P. 1975.On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4)103, 99–108. MR:54, # 113.Google Scholar
39. Erdös, P. 1980.Some combinatorial problems in geometry. Geometry and Differential Geometry (Proc. Conf. Univ. Haifa 1979), 46–53. (Lecture Notes in Math., Vol. 792). Springer, Berlin.Google Scholar
40. Erdös, P. 1982.Personal reminiscences and remarks on the mathematical work of Tibor Gallai. Combinatorica2, 207–212.Google Scholar
41. Erdös, P. 1983.Combinatorial problems in geometry. Math. Chronicle12, 35–54.Google Scholar
42. Erdös, P. 1984a.Some old and new problems on combinatorial geometry. Convexity and Graph Theory (Jerusalem, 1981), 129–136. (North-Holland Math. Studies, Vol. 87). MR:87b, # 52018.Google Scholar
43. Erdös, P. 1984b.Some old and new problems on combinatorial geometry. Annals Discrete Math.20, 129–136.Google Scholar
44. Erdös, P. 1984c.Research problems. Period Math. Hung.15, 101–103, Zbl 537.0515.Google Scholar
45. Erdös, P. 1985.Problems and results in combinatorial geometry. Discrete geometry and convexity (New York, 1982), 1–11. (Ann. New York Acad. Sci., Vol. 440). MR:87g, # 52001.Google Scholar
46. Erdös, P. andPurdy, G. 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, Vol. XVII). Utilitas Math., Winnipeg, Man. MR:55, # 10292.Google Scholar
47. Gallai, T. 1944.Solution to Problem 4065. Amer. Math. Monthly51, 169–171.Google Scholar
48. Grünbaum, B. 1956.A generalization of a problem of Sylvester. Riveon Lematematika10, 46–48. (Hebrew. English summary) MR:19, p. 667.Google Scholar
49. Grünbaum, B. 1972.Arrangements and Spreads. (CBMS, No. 10). Amer. Math. Soc., Providence, RI. MR:46, # 6148.Google Scholar
50. Grünbaum, B. 1975.Arrangements of colored lines. Notices Amer. Math. Soc.22, A-200.Google Scholar
51. Grünbaum, B. 1976.New views on some old questions of combinatorial geometry. (Colloq. Int. Theorie Comb., Roma 1973, Tomo 1), 451–468. MR:57, # 10605. Zbl 347.52004.Google Scholar
52. Grünbaum, B. andShephard, G. C. 1984.Simplicial arrangements in projective 3-space. (Coxeter Festschrift) Mitt. Math. Sem. Giessen166, 49–101.Google Scholar
53. Guy, R. K. 1971.Problem 15. Unsolved combinatorial problems. Combinatorial Mathematics and its Applications (Proceedings Conf. Oxford 1969, Editor, D. J. A. Welsh), 121–127. Academic Press, London. MR:43, # 3126.Google Scholar
54. Hadwiger, H., Debrunner, H. andKlee, V. 1964.Combinatorial Geometry in the Plane. Holt, Rinehart and Winston, New York. MR:29, # 1577.Google Scholar
55. Hanani, H. 1951.On the number of straight lines determined by n points. Riveon Lematematika5, 10–11. MR:13, p. 5.Google Scholar
56. Hanani, H. 1954/5.On the number of lines and planes determined by d points. Technion Israel Tech. Sci. Publ.6, 58–63.Google Scholar
57. Hansen, S. 1965.A generalization of a theorem of Sylvester on the lines determined by a finite point set. Math. Scand.16, 175–180. MR:34, # 3411.Google Scholar
58. Hansen, S. 1980.On configurations in 3-space without elementary planes and on the number of ordinary planes. Math. Scand.47, 181–194. Zbl 438.51002.Google Scholar
59. Hansen, S. 1981.Contributions to the Sylvester—Gallai theory. Ph.D. Thesis. Copenhagen.Google Scholar
60. Herzog, F. andKelly, L. M. 1960.A generalization of the theorem of Sylvester. Proc. Amer. Math. Soc.11, 327–331. MR:22, # 4044.Google Scholar
61. Kárteszi, F. 1963.Alcuni problemi della geometria d'incindenza. (Conf. Sem. Math. Univ. Bari, No. 88). FM:31, # 3926.Google Scholar
62. Kelly, L. M. 1986.A resolution of the Sylvester—Gallai problem of J.-P. Serre. Discrete Comput. Geom.1, 101–104.Google Scholar
63. Kelly, L. M. andMoser, W. 1958.On the number of ordinary lines determined by n points. Canad. J. Math.10, 210–219. MR:20, # 3494.Google Scholar
64. Kelly, L. M. andRottenberg, R. 1972.Simple points on pseudoline arrangements. Pacific J. Math.40, 617–622. MR:46, # 6150.Google Scholar
65. Lang, G. D. W. 1955.The dual of a well-known theorem. Math. Gazette39, 314.Google Scholar
66. Melchior, E. 1940.Über Vielseite der projektiven Ebene. Deutsche Math.5, 461–475. MR:3, p. 13.Google Scholar
67. Meyer, W. 1974.On ordinary points in arrangements. Israel J. Math.17, 124–135. MR:52, # 148.Google Scholar
68. Moser, W. 1957.Abstract Groups and Geometrical Configurations. Ph.D. Thesis. Univ. of Toronto.Google Scholar
69. Moser, W. 1961.An enuneration of the five parallelohedra. Cand. Math. Bull.4, 45–47. MR24a, # 3556.Google Scholar
70. Motzkin, T. 1951.The lines and planes connecting the points of a finite set. Trans. Amer. Math. Soc.70, 451–464. MR:12, p. 849.Google Scholar
71. Motzkin, T. 1967.Nonmixed connecting lines. Abstract 67T–605. Notices Amer. Math. Soc.14, 837.Google Scholar
72. Motzkin, T. 1975.Sets for which no point lies on many connecting lines. J. Combinatorial TheoryA 18, 345–348. MR51, # 2946.
73. Rottenberg, R. 1971.On finite sets of points in P 3. Israel J. Math.10, 160–171. MR:45, # 5879.Google Scholar
74. Rottenberg, R. 1973.Exceptional sets in a projective plane. (Technion Preprint Series, No. MT-169). Technion, Haifa.Google Scholar
75. Rottenberg, R. 1981.Visibility and parallelotopes in projective space. J. Geometry16, 19–29. MR:83b, 51005. Zbl 473.51012.
76. Rottenberg, R. 1982.Singular points in planar finite sets I. Geometriae Dedicata12, 407–415. Zbl 492.05021.
77. Sah, Chih-Han. 1986.The rich line problem of P. Erdös. (Colloquia Math. Soc. János Bolyai, Vol. 48. Intuitive Geometry, Siófok, 1985), 123–125.Google Scholar
78. Salamon, P. andErdös, P. 1988.The solution to a problem of Grünbaum. Canad. Math. Bull.31, 129–138.Google Scholar
79. Shannon, R. 1974. Ph.D. Thesis. Univ. Washington, Seattle.Google Scholar
80. Steinberg, R. 1944.Three point collinearity. Amer. Math. Monthly51, 169–171.
81. Sylvester, J. J. 1893.Mathematical Question 11851. Educational Times59, 98.Google Scholar
82. Szemerédi, E. andTrotter Jr., W. T. 1983a.A combinatorial distinction between the Euclidean and projective plane. European J. Combin.4, 385–394. Zbl 539.05026.Google Scholar
83. Szemerédi, E. andTrotter Jr., W. T. 1983b.Recent progress on extremal problems in discrete geometry. (Graphs and other combinatorial topics, Prague 1982). (Teubner-Text zur Math. Vol. 59), 316–319. Teubner, Leipzig.Google Scholar
84. Szemerédi, E. andTrotter Jr., W. T. 1983c.Extremal problems in discrete geometry. Combinatorica3, 381–392. Zbl 541.05012.Google Scholar
85. Tingley, D. 1975.Monochromatic lines in the plane. Math. Mag.48, 271–274, MR:53, # 9115.Google Scholar
86. Williams, V. C. 1968.A proof of Sylvester's theorem on collinear points. Amer. Math Monthly.75, 980–982.
87. Watson, K. S. 1981.Sylvester's problem for spreads of curves. Canad. J. Math.32, 219–239, Zbl 436. 51010.Google Scholar
88. Woodall, D. R. 1969.The λ-μ problem. J. Lond. Math. Soc. (2)1, 509–519.Google Scholar