Dense Arrangements are Locally Very Dense. I

Abstract

The Szemerédi–Trotter theorem [Combinatorica, 3 (1983), pp. 381–392] gives a bound on the maximum number of incidences between points and lines on the Euclidean plane. In particular it says that n lines and n points determine $O(n^{4/3})$ incidences. Let us suppose that an arrangement of n lines and n points defines $cn^{4/3}$ incidences, for a given positive c. It is widely believed that such arrangements have special structure, but no results are known in this direction. Here we show that for any natural number, k, one can find k points of the arrangement in general position such that any pair of them is incident to a line from the arrangement, provided by $n\geq n_0(k)$. In a subsequent paper we will establish a similar statement for hyperplanes.