A Combinatorial Distinction Between the Euclidean and Projective Planes

Let nand m be integers with n = m 2 + m + 1. Then the projective plane of order m has n points and" lines with each line containing m + 1 ""n 1/2 points. In this paper, we consider the analogous problem for the Euclidean plane and show that there cannot be a comparably large collection of lines each of which contains approximately n 1/ 2 points from a given set of n points. More precisely, we show that for every S > 0, there exist constants e, no so that if n ~ "0, it is not possible to find n points in the Euclidean plane and a collection of at least en 1/2 lines each containing at least Sn 1/2 of the points. This theorem answers a question posed by P. Erdos. The proof involves a covering lemma, which may be of independent interest, and an application of the first author's regularity lemma.