Fractional Sylvester-Gallai theorems
نویسندگان
چکیده
منابع مشابه
Fractional Sylvester-Gallai Theorems∗
We prove fractional analogs of the classical Sylvester-Gallai theorem. Our theorems translate local information about collinear triples in a set of points into global bounds on the dimension of the set. Specifically, we show that if for every points v in a finite set V ⊂ C, there are at least δ|V | other points u ∈ V for which the line through v, u contains a third point in V , then V resides i...
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We study questions in incidence geometry where the precise position of points is ‘blurry’ (e.g. due to noise, inaccuracy or error). Thus lines are replaced by narrow tubes, and more generally affine subspaces are replaced by their small neighborhood. We show that the presence of a sufficiently large number of approximately collinear triples in a set of points in C implies that the points are cl...
متن کاملSylvester-Gallai Theorems for Complex Numbers and Quaternions
A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai Theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a d...
متن کاملObservations and Computations in Sylvester-Gallai Theory
We bring together several new results related to the classical Sylvester-Gallai Theorem and its recently noted sharp dual. In 1951 Dirac and Motzkin conjectured that a configuration of n not all collinear points must admit at least n/2 ordinary connecting lines. There are two known counterexamples, when n = 7 and n = 13. We provide a construction that yields both counterexamples and show that t...
متن کاملThe Sylvester-Gallai theorem, colourings and algebra
Our point of departure is the following simple common generalisation of the Sylvester-Gallai theorem and the Motzkin-Rabin theorem: Let S be a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct points A, B ∈ S sharing a colour there is a third point C ∈ S, of the other colour, collinear with A and B. Then all the point...
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ژورنال
عنوان ژورنال: Proceedings of the National Academy of Sciences
سال: 2012
ISSN: 0027-8424,1091-6490
DOI: 10.1073/pnas.1203737109