Some Combinatorial Problems in the Plane

1 Let there be given n points in the plane. Denote by t i the number of lines which contain exactly i of the points (2 < i < n). The properties of the set {ti } have been studied a great deal. For example, there is the classical result of Gallai and Sylvester : Assume to = 0 (i .e ., the points are not all on one line) ; then t2 > 0. For the history of this problem see, e.g ., Motzkin [6] and Erdős [3, 4]. 1n this note we prove that some new and perhaps unexpected properties of the family {ti } hold. Let there be given n distinct points in the plane, not all on a line. We conjectured that for n > n o there always is an i such that t i > n-1. Krier and Straus pointed out that for n = 6 and 9 there are counterexamples. For n = 9, take the vertices of a square and its center and the four points of infinity determined by the sides and diagonals of the square. For n = 13 we also get a counterexample from the vertices of a regular hexagon and its center and the six points of infinity determined by the sides and diagonals of the hexagon. Nevertheless, the conjecture is true for n , 25. In fact we show in Theorem 1 that we can always choose i = 2 or 3, and that max ti = max(t2 , t 3). i Assume now that no line has more than (1-e)q points on it. Then we are convinced that there is an 7) = 7)(e) such that