NDLI: Dense Arrangements are Locally Very Dense. I

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Solymosi, Jzsef
Copyright Year	2006
Abstract	The SzemerdiTrotter theorem Combinatorica, 3 1983, pp. 381392 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.
Starting Page	623
Ending Page	627
Page Count	5
File Format	PDF
ISSN	08954801
DOI	10.1137/05062826X
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	3
Volume Number	20
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-08-29
Access Restriction	Subscribed
Subject Keyword	regularity lemma SzemerdiTrotter theorem Erdős problems and related topics of discrete geometry pointline incidences Combinatorial complexity of geometric structures Planar arrangements of lines and pseudolines
Content Type	Text
Resource Type	Article
Subject	Mathematics

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