Feature Column

A Discrete Geometrical Gem

5. References

Aigner, M. and G. Ziegler, Proofs From the Book, Springer-Verlag, Berlin, 1998.
Baston, V. and F. Bostick, A Gallai-type problem, J. Combin. Theory, Ser. A, 24 (1978) 122-125.

Alon, N. and H. Last, R. Pinchasi, and M. Sharir, On the complexity of arrangements of circles in the plane, Discrete Comput. Geom., 26 (2001) 465-492.

Bálint, V., On a connection between unit circles and horocycles determined by n points, Periodica Math. Hung., 38 (1999) 15-17.

Bálint, V., A short survey of (r,q)-structures, Discrete Comput. Geometry, to appear.

Bálintová, A. and V. Bálint, On the number of circles determined by n points in the Euclidean plane, Discrete Comput. Geom., 26 (2001) 465-492.

Bezdek, A., On unit circles which avoid all but two points of a given point-set, European J. Combin., 23 (2002) 11-13.

Bezdek, A. and F. Fodor, I. Talata, Sylvester-type theorems for unit circles, Discrete Math., 241 (2001) 97-101.

Bokowski, J. and J. Richter-Gebert, A new Sylvester-Gallai configuration representing the 13-point projective plane in R4, J. Combinatorial Theory Ser. B 54 (1992) 161-165.

Bonnice, W. and M. Edelstein, Flats associated with finite sets in Pd, Nieuw Arch. Wisk., 15 (1967) 11-14.

Bonnice, W. and L. Kelly, On the number of ordinary planes, J. Comb. Theory 11 (1971) 45-53.

Boros, E., and Z. Füredi, L. Kelly, On representing Sylvester-Gallai designs, Discrete Computational Geometry, 4 (1989) 345-348.

Borwein, P., Monochrome lines in the plane, Math. Mag., 52 (1979) 41-45.

Borwein, P., On monochrome lines and hyperplanes, J. Comb. Theory, A 33 (1982) 76-81.

Borwein, P., On Sylvester's problem and Haar spaces, Pacific J. Math., 109 (1983) 275-278.

Borwein, P. Sylvester's problem and Motzkin's theorem for countable compact sets, Proc. Amer. Math. Soc., 90 (1984) 580-584.

Borwein, P. and M. Edelstein, A conjecture related to Sylvester's problem, Amer. Math. Monthly, 90 (1983) 389-390.
Borwien, P. and W. Moser, A survey of Sylvester's problem and its generalizations, Aequationes Mathematicae, 40 (1990) 111-135.

Chakerian, G., Sylvester's problem on collinear points and a relative, Amer. Math. Monthly, 77 (1970) 164-167.

Chvátal, V., Sylvester-Gallai theorem and metric betweenness, to appear, Disc. Comput. Geometry. (Available on the web.)

Coxeter, H., A problem of collinear points, Amer. Math. Monthly, 55 (1948) 26-28.

Coxeter, H., Introduction to Geometry, 2nd. edition, Wiley, New York, 1969.

Crowe, D. and T. McKee, Sylvester's problem on collinear points, Math. Mag., 41 (1968) 30-34.

Csima, J. and E. Sawyer, There exist 6n/13 ordinary points, Discrete and Compt. Geom., 9 (1993) 187-202.

Csima, J. and E. Sawyer, The 6n/13 theorem revisited, in Proc. Seventh Inter. Conf. in Graph Theory, Combinatorics, Algorithms, and Applications, Y. Alavi and A. Schenk (eds.), Wiley, New York, p. 235-249.

da Sivla, P. and K. Fukuda, Isolating points by lines in the plane, J. Geom., 62 (1998) 48-65.

Dirac, G., Collinearity properties of sets of points, Quarterly J. Math., 2 (1951) 221-227.

Edelstein, M., A further generalization of a problem of Sylvester, Riveon Lamatematka, 11 (1957) 50-55.

Edelstein, M. Generalizations of the Sylvester problem, Math. Mag., 43 (1970) 250-254.

Edelstein, M. and F. Herzog, L. Kelly, A further theorem of the Sylvester type, Proc. Amer. Math. Soc., 14 (1963) 359-363.

Elekes, G., n point in the plane can determine n3/2 unit circles, Combinatorica, 4 (1984) 131.

Elkies, N., The planarity of Sylvester-Gallai configurations over C. (Preprint.)

Erdös, P., Problem 4065, Amer. Math. Monthly, 50 (1944) 65.

Erdös, P., Personal reminiscences and remarks of other mathematical work of Tibor Gallai, Combinatorica, 2 (1982) 207-212.

Finschi, L., A Graph Theoretical Approach for Reconstruction and Generation of Oriented Matroids, Ph. D. Thesis, Swiss Federal Institute of Technology, 2001.

Finschi, L. and K. Fukuda, Complete combinatorial generation of small point configurations and hyperplane arrangements, (Preprint.)

Finschi, L. and K. Fukuda, Generation of oriented matroids - a graph theoretical approach, to appear, Discrete Comput. Geom.

Gallai (Grünwald), T., Solution to problem 4065, Amer. Math. Monthly, 51 (1944)169-171.

Goodman, J. and J. O'Rourke, (eds.), Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton, 1997.

Goodman, J. and R. Pollack, Multidimensional sorting, Siam J. Computing, 12 (1983) 484-507.

Grünbaum, B., A generalization of a problem of Sylvester, Riveon Lematematika, 10 (1956) 667.

Grünbaum, B., Arrangements and Spreads, American Mathematical Society, Providence, 1972.

Grünbaum, B., Arrangements of hyperplanes, in Proceedings of the Second Louisiana Conference on Combinatorics,Graph Theory and Computing, R. Mullin, K. Reid, D. Roselle, and R. Thomas, (eds.), Louisiana State U. Baton Rouge, 1971, p. 41-106.

Grünbaum, B., Arrangements of colored lines, Notices Amer. Math. Soc., 22 (1975) A-200.

Hansen, S., A generalization of a theorem of Sylvester on the lines determined by a finite point set, Math. Scand., 47 (1965) 175-180.

Hansen, S., On configurations in 3-space without elementary planes and on the number of ordinary planes, Math. Scand., 47 (1980) 181-194.

Hansen, S., Contributions to the Sylvester-Gallai Theory, Ph.D. Thesis, University of Copenhagen, 1981.

Kelly, L. A resolution of the Sylvester-Gallai problem of J.-P. Serre, Discrete Comput. Geom., 1986 (1) 101-104.

Kelly, L. and W. Moser, On the number of ordinary lines determined by n points, Canadian J. Math., 10 (1958) 210-219.

Kelly, L. and S. Nwankpa, Affine embeddings of Sylvester-Gallai designs, J. Combinatorial Theory Ser. A 14 (1973) 422-438.

Klee, V. and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, Mathematical Association of America, Washington, 1991.

Kupitz, Y., On a generalization of the Gallai-Sylvester theorem, Discrete Comput. Geom., 7 (1992) 87-103.

Lin, X., Another brief proof of Sylvester Theorem, Amer. Math. Monthly, 95 (1988) 932-933.

Melchior, E., Über Vielseite der projektiven Ebene, Deutsche Mathematik, 5 (1940) 461-475.

Meyer, W., On ordinary points in arrangements, Israel J. Math., 17 (1974) 124-135.

Motzkin, T., The lines and planes connecting the points of a finite set, Trans. Amer. Math. Soc., 70 (1951) 451-464.

Motzkin, T., Sets for which no point lies on many connecting lines, J. Combinatorial Theory A 18 (1975) 345-348.

Mukhopadhyay, A. and A. Agrawal, R. Hosabettu, On the ordinary line problem in computational geometry, Nordic J. Comput., 4 (1997) 330-341.

Nevo, E. and J. Pach, R. Pinchasi, and M. Sharir, Lenses in arrangements of pseudodisks and their application. (Preprint.)

Pach, J. and P. Agarwal, Combinatorial Geometry, Wiley, New York, 1995.

Pach, J. and R. Pinchasi, Bichromatic lines with few points, J. Combin. Theory Ser. A, 90 (2000) 326-335.

Palásti, I., A construction for arrangements of lines with vertices of large multiplicity, Stud. Sci. Math. Hung., 21 (1986) 67-78.

Pinchasi, R., Gallai-Sylvester theorem for pairwise intersecting unit circles, Discrete and Computational Geometry 28 (1902) 607-624.

Pretorius, L. and K. Swanepoel, An algorithmic proof of the Motzkin-Rabin theorem on monochrome lines. (Preprint.)

Rottenberg, R., On finite sets of points in P3, Israel J. Math., 10 (1971) 160-171.

Sylvester, J., Mathematical question 11851, Educational Times, 59 (1893) 98.

Watson, K., Sylvester's problem for spreads of curves, Canad. J. Math., 32 (1981) 219-239.

Wiseman, J. and P. Wilson, A Sylvester theorem for conic sections, Discrete Comp. Geom., 3 (1988) 295-305.

Yang, Y., On a conjecture about monochromatic flats, Discrete Math., 80 (1990) 213-216.

Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials.

  1. Introduction
  2. Planar point configurations
  3. Some ramifications of the Sylvester-Gallai Theorem
  4. Generalizations and adding color
  5. References

Welcome to the
Feature Column!

These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.
Read more . . .

Search Feature Column

Feature Column at a glance

Show Archive

Browse subjects

American Mathematical Society