On the Number of Ordinary Lines Determined by n Points

Abstract

More than sixty years ago, Sylvester (13) proposed the following problem: Let n given points have the property that the straight line joining any two of them passes through a third point of the set. Must the n points all lie on one line? An alleged solution (not by Sylvester) advanced at the time proved to be fallacious and the problem remained unsolved until about 1933 when it was revived by Erdös (7) and others.

On Sets Defining Few Ordinary Lines

“…Kelly and Moser [23] observe that a triangle together with the midpoints of its sides and its centroid has n = 7 and just 3 ordinary lines. Crowe and McKee [10] provide a more complicated configuration with n = 13 and 6 ordinary lines.…”

“…It is natural to wonder how many ordinary lines there are in a set of P points, not all on a line, when the cardinality |P | of P is equal to n. Melchior's argument in fact shows that there are at least three ordinary lines, but considerably more is known. Motzkin [27] was the first to obtain a lower bound (of order n 1/2 ) tending to infinity with n. Kelly and Moser [23] proved that there are at least 3n/7 ordinary lines, and Csima and Sawyer [11] improved this to 6n/13 when n > 7. Their work used ideas from the thesis of Hansen [20], which purported to prove the n/2 lower bound but was apparently flawed.…”

There Are Not Too Many Magic Configurations

“…First of all, we can assume that |A| ≥ 3 in Theorem 2, and hence, by duality, also in Theorem 3: any noncollinear set of points determines at least 3 ordinary lines (see [KM58]), and this would be We consider the arrangement B of the circles in B on the sphere S. Observe that any crossing point on a circle b ∈ B, even with a circle in A, is a crossing point in B. Indeed, otherwise either it is an ordinary intersection point on b, or it is an intersection point that is not ordinary of at least two circles in A, contrary to our assumptions.…”

“…By the theorem of Kelly and Moser [KM58], the set P \{p} determines at least 3 7 (n−1) ordinary lines (see [CS93] for the current best bound on the number of ordinary lines determined by n points). By our assumption, all these lines must pass through p. It follows that at most 1 7 (n − 1) points of P \ {p} lie on an ordinary line through p and these are all the ordinary lines determined by P , contradicting the Kelly-Moser theorem.…”

On Zonotopes

“…Many results about zonotopes have been deduced from consideration of the corresponding arrangements (see, for example, Coxeter [1962] and Kelly-Moser [1958]), but there are few, if any, results in the other direction. Here, however, this situation can be remedied.…”