Let n and m be integers with n = m2 + m + 1. Then the projective plane of order m has n points and n lines with each line containing m + ≈ n1/2 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 n1/2 points from a given set of n points. More precisely, we show that for every δ 〉 0, there exist constants c, n0 so that if n ⩾ n0, it is not possible to find n points in the Euclidean plane and a collection of at least cn1/2 lines each containing at least δn1/2 of the points. This theorem answers a question posed by P. Erdös. The proof involves a covering lemma, which may be of independent interest, and an application of the first author's regularity lemma.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics