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A003034 Sylvester's problem: minimal number of ordinary lines through n points in the plane.
(Formerly M2275)
0
3, 3, 4, 3, 3, 4, 6, 5, 6, 6, 6, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

An ordinary line contains exactly 2 points. The problem is to place n points, not all on a line, so as to minimize the number of ordinary lines.

Pach and Sharir give the following table for n >= 3: 3, 3, 4, 3, 3, 4, 6, 5, 6, 6, 6, 7, ?, 8, ?, 9, ?, ?, ?, 22, ... - N. J. A. Sloane, Dec 01 2010

REFERENCES

J. Brakke, Some new values for Sylvester's function for n collinear points, J. Undergrad. Math., 4 (1972), 11-14.

H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, F12.

B. Gr├╝nbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 18.

S. Hansen, Contributions to the Sylvester-Gallai theory, Dissertation, Univ. Copenhagen, 1981.

J. Pach and M. Sharir, Combinatorial Geometry and Its Algorithmic Applications, Amer. Math. Soc., 2009; see p. 2.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=3..14.

P. Borwein and W. O. J. Moser, A survey of Sylvester's problem and its generalizations, Aequat. Math., 40 (1990), 111-135.

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

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

FORMULA

Kelly and Moser showed that a(n) >= ceiling(3n/7); Hansen showed that a(n) >= floor(n/2) except for n=7 and 13.

CROSSREFS

Sequence in context: A188590 A080038 A121937 * A091282 A202708 A027684

Adjacent sequences:  A003031 A003032 A003033 * A003035 A003036 A003037

KEYWORD

nonn,hard,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 29 10:34 EST 2020. Contains 331337 sequences. (Running on oeis4.)