Block configurations on real cubic curves

Abstract

A 3 k-block is a set of 3 k points on a real non-singular cubic curve, which are the points of intersection of a real cubic curve and a curve of decree k. It has been proved [3]that 3 k -I points on a real cubic curve uniquely determine- the remaining point of a 3k-block and that this point can be located using only straight edge constructions. An n 3 k configuration consists of n points and n 3k-subsets, such that any two of the n points are on at most I of the subsets (we abbreviate 3 k-subset to subset when confusion cannot result) and any, two of the subsets have at most 1 point in common. It is proved that such n 3 k configurations exist with the n points on a cubic curve and the 3k-subsets being 3k-blocks of points on the cubic.