Dense arrangements are locally very dense I

Abstract

The Szemer\'edi-Trotter theorem 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 similar statement to hyperplanes.