Extremal problems in discrete geometry

Begin with a set of four points in the real plane in general position. Add to this collection the intersection of all lines through pairs of these points. Iterate. Ismailescu and Radoičić (2003) showed that the limiting set is dense in the plane. We give doubly exponential upper and lower bounds on the number of points at each stage. The proof employs a variant of the Szemerédi-Trotter Theorem and an analysis of the "minimum degree" of the growing configuration.Consider the iterative process of constructing points and lines in the real plane given by the following: begin with a set of points P 1 = {p 1 , p 2 , p 3 , p 4 } in the real plane in general position. For each pair of points, construct the line passing through the pair. This will create a set of lines L 1 = {ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 , ℓ 5 , ℓ 6 }. Some of these constructed lines will intersect at points in the plane that do not belong to the set P 1 . Add any such point to the set P 1 to get a new set P 2 . Now, note that there exist some pairs of points in P 2 that do not lie on a line in L 1 , namely some elements of P 2 \ P 1 . Add these missing lines to the set L 1 to get a new set L 2 . Iterate in this manner, adding points to P k followed by adding lines to L k . We assume that the original configuration is such that for every k ∈ N no two lines in L k are parallel. Now we introduce some notation for this iterative process. The k th stage is defined to consist of these two ordered steps:1. Add each intersection of pairs of elements of L k to P k+1 , and 2. Add a line through each of pair of elements of P k to L k+1 .

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“…The argument closely resembles Székely's proof ( [4]) of the Szemerédi-Trotter Theorem (first appearing in [5]). …”

supporting

confidence: 62%

Iterated Point–Line Configurations Grow Doubly-Exponentially

Cooper

Walters

2009

Discrete Comput Geom

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“…One of the objectives in combinatorial incidence geometry is to obtain good bounds on the cardinality |I(P, L)| on the number of incidences between finite collections P, L, subject to various hypotheses on P and L. For instance, we have the classical result of Szemerédi and Trotter [48]: Theorem 1.1 (Szemerédi-Trotter theorem). [48] Let P be a finite set of points in R for some absolute constant C.…”

Section: Introductionmentioning

confidence: 99%

An Incidence Theorem in Higher Dimensions

Solymosi

Tao²

2012

Discrete Comput Geom

114

143

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“…Such bounds, apart from their intrinsic interest, have found applications to a variety of other combinatorial problems, e.g., [3], [5], [23], [24]; a comprehensive survey of the area is given in [17]. A prototype result here is the classical Szemerédi-Trotter incidence theorem [29], which states that the total number of incidences I between N points and M straight lines in the plane obeys (1.1) I N + M + (NM) There are explicit examples showing that this bound is sharp. Here and below, c, C are constants; X Y means that X ≤ CY for some C, and | · | denotes the cardinality of a finite set, the Euclidean norm of a vector in R d , or the absolute value of a complex number, depending on the context.…”

Section: Introductionmentioning

confidence: 99%