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%I M2275 #25 Jan 01 2022 08:07:27
%S 3,3,4,3,3,4,6,5,6,6,6,7
%N Sylvester's problem: minimal number of ordinary lines through n points in the plane.
%C 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.
%C Pach and Sharir give the following table for n >= 3: 3, 3, 4, 3, 3, 4, 6, 5, 6, 6, 6, 7, ?, 8, ?, 9, ?, ?, ?, 11, ... - _N. J. A. Sloane_, Dec 01 2010
%D J. Brakke, Some new values for Sylvester's function for n collinear points, J. Undergrad. Math., 4 (1972), 11-14.
%D H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, F12.
%D B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 18.
%D S. Hansen, Contributions to the Sylvester-Gallai theory, Dissertation, Univ. Copenhagen, 1981. [Csima and Sawyer point out errors in this dissertation.]
%D J. Pach and M. Sharir, Combinatorial Geometry and Its Algorithmic Applications, Amer. Math. Soc., 2009; see p. 2.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H P. Borwein and W. O. J. Moser, <a href="http://dx.doi.org/10.1007/BF02112289">A survey of Sylvester's problem and its generalizations</a>, Aequat. Math., 40 (1990), 111-135.
%H D. W. Crowe and T. A. McKee, <a href="http://www.jstor.org/stable/2687957">Sylvester's problem on collinear points</a>, Math. Mag., 41 (1968), 30-34.
%H J. Csima and E. T. Sawyer, <a href="https://doi.org/10.1007/BF02189318">There exist 6n/13 ordinary points</a>, Discrete & Computational Geometry, 9 (1993), 187-202.
%H L. M. Kelley and W. O. J. Moser, <a href="http://dx.doi.org/10.4153/CJM-1958-024-6">On the number of ordinary lines determined by n points</a>, Canad. J. Math., 10 (1958), 210-219.
%F Kelly and Moser showed that a(n) >= ceiling(3n/7); Csima and Sawyer showed that a(n) >= floor(6n/13) except for n=7.
%K nonn,hard,nice
%O 3,1
%A _N. J. A. Sloane_
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