Extremal problems in discrete geometry

Abstract

In this paper, we establish several theorems involving configurations of points and lines in the Euclidean plane. Our results answer questions and settle conjectures of P. Erdõs, G. Purdy, and G. Dirac. The principal result is that there exists an absolute constantc ₁ so that when$\sqrt n \leqq t \leqq \left( {_2^n } \right)$, the number of incidences betweenn points andt lines is less thanc ₁ n ^2/3 t ^2/3. Using this result, it follows immediately that there exists an absolute constantc ₂ so that ifk≦√n, then the number of lines containing at leastk points is less thanc ₂ n ²/k ³. We then prove that there exists an absolute constantc ₃ so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc ₃ n of the lines determined by then points. Finally, we show that there is an absolute constantc ₄ so that there are less than exp (c ₄ √n) sequences 2≦y ₁≦y ₂≦...≦y _r for which there is a set ofn points and a setl ₁,l ₂, ...,l _t oft lines so thatl _j containsy _j points.