A survey of Sylvester's problem and its generalizations

Borwein, Peter; Moser, W. O. J.

doi:10.1007/bf02112289

Cited by 77 publications

(56 citation statements)

References 50 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The question is thus whether there exists a set of points such that through each point there are many lines that contain many other points. This question reminds us of the well-known Sylvester-Gallai (SG) theorem saying that for every set of points in the real plane, not all on the same line, there is a line passing through exactly two points (see Borwein et al 1990 for a survey of this theorem and its generalizations). To demonstrate the connection to this theorem, consider the question of constructing a two-query LCC that is "perfect" in the sense that for each coordinate, the set of pairs that are used to decode it covers the set of remaining coordinates perfectly (in particular, there is an odd number of coordinates).…”

Section: Lccs Over the Reals And The Sylvester-gallai Theoremmentioning

confidence: 99%

On Matrix Rigidity and Locally Self-correctable Codes

Dvir

2011

comput. complex.

View full text Add to dashboard Cite

We describe a new connection between the problem of finding rigid matrices, as posed by Valiant (MFCS 197 ), and the problem of proving lower bounds for linear locally correctable codes. Our result shows that proving linear lower bounds on locally correctable codes with super-logarithmic query complexity will give new constructions of rigid matrices. The interest in constructing rigid matrices is their connection to circuit lower bounds. Our results are based on a lemma saying that if the generating matrix of a locally decodable code is not rigid, then it defines a locally selfcorrectable code with rate close to one. Thus, showing that such codes cannot exist will prove that the generating matrix of any locally decodable code (and in particular Reed Muller codes) is rigid. This connection gives, on the one hand, a new approach to attack the long-standing open problem of matrix rigidity and, on the other hand, explains the difficulty of advancing our current knowledge on locally correctable codes (in the high-query regime).

show abstract

Section: Lccs Over the Reals And The Sylvester-gallai Theoremmentioning

confidence: 99%

On Matrix Rigidity and Locally Self-correctable Codes

Dvir

2011

comput. complex.

View full text Add to dashboard Cite

show abstract

“…4. We are also grateful to Noga Alon, Boris Bukh, Frando Mariacci, Bjorn Poonen and Jozsef Solymosi for helpful comments and conversations, and to Frank de Zeeuw for pointing out the case of the acnodal singular cubic curve, which was not treated properly in the first draft of this paper.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…An illuminating discussion of this point may be found in the MathSciNet review of [11]. There are several nice surveys on this and related problems; see [4], [15,Chapter 17], [28] or [29].…”

Section: Introductionmentioning

confidence: 99%

On Sets Defining Few Ordinary Lines

Green¹,

Tao²

2013

Discrete Comput Geom

167

View full text Add to dashboard Cite

Abstract. Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P . This confirms, for large n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than n − C ordinary lines for some absolute constant C. We also solve, for large n, the "orchard-planting problem", which asks for the maximum number of lines through exactly 3 points of P . Underlying these results is a structure theorem which states that if P has at most Kn ordinary lines then all but O(K) points of P lie on a cubic curve, if n is sufficiently large depending on K.

show abstract

“…Later it resurfaced as a conjecture by Paul Erdös [2]: If a finite set of points in the plane is not on one line then there is a line through exactly two of the points. Since then there has appeared a substantial literature (seen in [1]) on the problem and its generalizations. For example, in [5] Motzkin considered n points in the plane, not all on a line and not all on a circle, and showed that there is either a circle or a line containing exactly three of the points.…”

Section: Introductionmentioning

confidence: 99%