Abstract
Try for yourself before you read much further to construct configurations of points in the plane that determine “relatively few” slopes. For this we assume, of course, that the n > 3 points do not all lie on one line. Recall from Chapter 9 on “Lines in the plane” the theorem of Erdös and de Bruijn: the n points will determine at least n different lines. But of course many of these lines may be parallel, and thus determine the same slope.
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References
J. E. Goodman & R. Pollack: A combinatorial perspective on some problems in geometry, Congressus Numerantium 32 (1981), 383–394.
R. E. Jamison & D. Hill: A catalogue of slope-critical configurations, Congressus Numerantium 40 (1983), 101–125.
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P. Ungar: 2N noncollinear points determine at least 2N directions, J. Combinatorial Theory, Ser. A 33 (1982), 343–347.
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© 2004 Springer-Verlag Berlin Heidelberg
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Aigner, M., Ziegler, G.M. (2004). The slope problem. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05412-3_10
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DOI: https://doi.org/10.1007/978-3-662-05412-3_10
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