Many Collinear k-Tuples with no k+1 Collinear Points

Abstract

For every k>3, we give a construction of planar point sets with many collinear k-tuples and no collinear (k+1)-tuples. We show that there are n0=n0(k) and c=c(k) such that if n≥ n0, then there exists a set of n points in the plane that does not contain k+1 points on a line, but it contains at least n2-(c√/ log n) collinear k-tuples of points. Thus, we significantly improve the previously best known lower bound for the largest number of collinear k-tuples in such a set, and get reasonably close to the trivial upper bound O(n2). © 2013 Springer Science+Business Media New York.