The slope problem

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 11 on “Lines in the plane” the theorem of Erdős 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.