On grids in point-line arrangements in the plane

Abstract

The famous Szemerédi-Trotter theorem states that any arrangement of n points and n lines in the plane determines O(n 4/3 ) incidences, and this bound is tight. In this paper, we prove the following Turán-type result for point-line incidence. Let L 1 and L 2 be two sets of t lines in the plane and let P = { 1 ∩ 2 : 1 ∈ L 1 , 2 ∈ L 2 } be the set of intersection points between L 1 and L 2 . We say that (P, L 1 ∪ L 2 ) forms a natural t × t grid if |P | = t 2 , and conv(P ) does not contain the intersection point of some two lines in L i , for i = 1, 2. For fixed t > 1, we show that any arrangement of n points and n lines in the plane that does not contain a natural t × t grid determines O(n 4 3 −ε ) incidences, where ε = ε(t). We also provide a construction of n points and n lines in the plane that does not contain a natural 2 × 2 grid and determines at least Ω(n 1+ 1 14 ) incidences.