Sylvester's Problem on Collinear Points

Crowe, Donald W.; McKee, Terry A.

doi:10.1080/0025570x.1968.11975829

Cited by 18 publications

(13 citation statements)

References 6 publications

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“…For n even (n ≥ 6), n/2 is the best possible lower bound for m 2 (n), because arrangements of n lines with exactly n/2 simple crossings are constructed by Böröczky (see [9]). The arrangements consist of n/2 lines containing the edges of a regular n/2-gon and together with n/2 lines of reflective symmetry.…”

Section: On the Number Of Ordinary Lines And Planesmentioning

confidence: 99%

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Matroid Enumeration for Incidence Geometry

Matsumoto

Moriyama

Imai

et al. 2011

Discrete Comput Geom

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Matroids are combinatorial abstractions for point configurations and hyperplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinearity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of incidence problem, the points-lines-planes conjecture and the so-called Sylvester-Gallai type problems derived from the Sylvester-Gallai theorem, by developing a new algorithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh-Seymour on ≤11 points in R 3 and that of Motzkin on ≤12 lines in R 2 , extending previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of oriented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n ≤ 12 elements and rank 4 on n ≤ 9 elements. We further list several new minimal non-orientable matroids.

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Section: On the Number Of Ordinary Lines And Planesmentioning

confidence: 99%

“…4 (together with the line at infinity, Fig. 4 is the "McKee configuration" of 13 lines with six simple crossings) [9].…”

Section: On the Number Of Ordinary Lines And Planesmentioning

confidence: 99%

Matroid Enumeration for Incidence Geometry

Matsumoto

Moriyama

Imai

et al. 2011

Discrete Comput Geom

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show abstract

“…Only two arrangements with fewer than n 2 ordinary points are known: the seven line arrangement due to Kelly and Moser and a 13 line arrangement due to McKee [3]. An infinite family of arrangements with n lines and exactly n 2 ordinary points is known and credited to Böröczky (see [2]).…”

Section: Discussionmentioning

confidence: 98%

On the affine Sylvester problem

Lenchner

2011

Discrete Applied Mathematics

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“…This is in agreement with the multiplicity of the corresponding point L in the dual projective plane since the points on L correspond to lines passing through L . Our description is borrowed from Crowe and McKee survey [9]. Dirac's conjecture has been proved in steps.…”

Section: Arrangements In the Real Projective Planementioning

confidence: 99%

A few introductory remarks on line arrangements

Szpond¹

2019

Analytic and Algebraic Geometry 3

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Points and lines can be regarded as the simplest geometrical objects. Incidence relations between them have been studied since ancient times. Strangely enough our knowledge of this area of mathematics is still far from being complete. In fact a number of interesting and apparently difficult conjectures has been raised just recently. Additionally a number of interesting connections to other branches of mathematics have been established. This is an attempt to record some of these recent developments. 2010 Mathematics Subject Classification. 14C20 and MSC 14N20 and MSC 13A15.

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Sylvester's Problem on Collinear Points

Cited by 18 publications

References 6 publications

Matroid Enumeration for Incidence Geometry

Matroid Enumeration for Incidence Geometry

On the affine Sylvester problem

A few introductory remarks on line arrangements

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