all sufficiently large point sets (that is, for some suitable choice of ), the number of ordinary lines is indeed at least . Furthermore, when is odd, the number of ordinary lines is at least , for some constant . Thus, the constructions of Böröczky for even and odd (discussed above) are best possible. Minimizing the number of ordinary lines is closely related to the orchard-planting problem of maximizing the number of three-point lines, which Green and Tao also solved for all sufficiently large point sets. The number of connecting lines As Paul Erdős observed, the Sylvester–Gallai theorem immediately implies that any set of points that are not collinear determines at least different lines. This result is known as the De Bruijn–Erdős theorem. As a base case, the result is clearly true for . For any larger value of , the result can be reduced from points to points, by deleting an ordinary line and one of the two points on it (taking care not to delete a point for which the remaining subset would lie on a single line). Thus, it follows by mathematical induction. The example of a near-pencil, a set of collinear points together with one additional point that is not on the same line as the other points, shows that this bound is tight. Generalizations The Sylvester–Gallai theorem has been generalized to colored point sets in the Euclidean plane, and to systems of points and lines defined algebraically or by distances in a metric space. In general, these variations of the theorem consider only finite sets of points, to avoid examples like the set of all points in the Euclidean plane, which does not have an ordinary line. Colored points A variation of Sylvester's problem, posed in the mid-1960s by Ronald Graham and popularized by Donald J. Newman, considers finite planar sets of points (not all in a line) that are given two colors, and asks whether every such set has a line through two or more points that are all the same color. In the language of sets and families of sets, an equivalent statement is that the family of the collinear subsets of a finite point set (not all on one line) cannot have Property B. A proof of this variation was announced by Theodore Motzkin but never published; the first published proof was by . Non-real coordinates Just as the Euclidean plane or