## 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 constant*c* _{1} so that when\(\sqrt n \leqq t \leqq \left( {_2^n } \right)\), the number of incidences between*n* points and*t* lines is less than*c* _{1} *n* ^{2/3} *t* ^{2/3}. Using this result, it follows immediately that there exists an absolute constant*c* _{2} so that if*k*≦√*n*, then the number of lines containing at least*k* points is less than*c* _{2} *n* ^{2}/*k* ^{3}. We then prove that there exists an absolute constant*c* _{3} so that whenever*n* points are placed in the plane not all on the same line, then there is one point on more than*c* _{3} *n* of the lines determined by the*n* points. Finally, we show that there is an absolute constant*c* _{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 of*n* points and a set*l* _{1},*l* _{2}, ...,*l* _{ t } of*t* lines so that*l* _{ j } contains*y* _{ j } points.

## AMS subject classification (1980)

51 M 05 05 C 35## Preview

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## References

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