whether approaches infinity with . confirmed that it does by proving that . conjectured that , for all values of , a conjecture that still stands . This is often referred to as the Dirac–Motzkin conjecture; see for example . proved that . Dirac's conjectured lower bound is asymptotically the best possible, as the even numbers greater than four have a matching upper bound . The construction, due to Károly Böröczky, that achieves this bound consists of the vertices of a regular -gon in the real projective plane and another points (thus, ) on the line at infinity corresponding to each of the directions determined by pairs of vertices. Although there are pairs of these points, they determine only distinct directions. This arrangement has only ordinary lines, the lines that connect a vertex with the point at infinity collinear with the two neighbors of . As with any finite configuration in the real projective plane, this construction can be perturbed so that all points are finite, without changing the number of ordinary lines. For odd , only two examples are known that match Dirac's lower bound conjecture, that is, with One example, by , consists of the vertices, edge midpoints, and centroid of an equilateral triangle; these seven points determine only three ordinary lines. The configuration in which these three ordinary lines are replaced by a single line cannot be realized in the Euclidean plane, but forms a finite projective space known as the Fano plane. Because of this connection, the Kelly–Moser example has also been called the non-Fano configuration. The other counterexample, due to McKee, consists of two regular pentagons joined edge-to-edge together with the midpoint of the shared edge and four points on the line at infinity in the projective plane; these 13 points have among them 6 ordinary lines. Modifications of Böröczky's construction lead to sets of odd numbers of points with ordinary lines. proved that except when is seven. Asymptotically, this formula is already of the proven upper bound. The case is an exception because otherwise the Kelly–Moser construction would be a counterexample; their construction shows that . However, were the Csima–Sawyer bound valid for , it would claim that . A closely related result is Beck's theorem, stating a tradeoff between the number of lines with few points and the number of points on a single line. Ben Green and Terence Tao showed that for