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54
SylvesterGallai theorem and metric betweenness
, 2002
"... Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines fro ..."
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Cited by 10 (2 self)
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from Euclidean spaces to arbitrary metric spaces. We present one of them and conjecture that, with lines in metric spaces defined in this way, the SylvesterGallai theorem generalizes as follows: in every finite metric space, there is a line consisting of either two points or all the points
Extremal Problems Related to the Sylvester–Gallai Theorem
"... Abstract. We discuss certain extremal problems in combinatorial geometry, including Sylvester’s problem and its generalizations. 1. ..."
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Abstract. We discuss certain extremal problems in combinatorial geometry, including Sylvester’s problem and its generalizations. 1.
From SylvesterGallai Congurations to Rank Bounds: Improved Blackbox Identity Test for Depth3 Circuits
"... We study the problem of identity testing for depth3 circuits of top fanin k and degree d. We give a new structure theorem for such identities that improves the known deterministic dk O(k)time blackbox identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes dO(k 2)time. Our s ..."
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dependent quantity, the SylvesterGallai rank bound, to the rank of depth3 identities. We also prove a high dimensional SylvesterGallai theorem for all elds, and get a general depth3 identity rank bound (slightly improving previous bounds).
The SylvesterChvatal Theorem
 Discrete & Computational Geometry
"... The SylvesterGallai theorem asserts that every finite set S of points in twodimensional Euclidean space includes two points, a and b, such that either there is no other point in S is on the line ab, or the line ab contains all the points in S.V.Chvatal extended the notion of lines to arbitrary ..."
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Cited by 7 (3 self)
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metric spaces and made a conjecture that generalizes the SylvesterGallai theorem. In the present article we prove this conjecture.
Some Problems in Discrete Geometry
, 2006
"... The SylvesterGallai theorem asserts that any noncollinear point set in the plane determines a line passing through exactly two points in the set. The problem was posed by Sylvester in 1893 and first solved by Gallai in 1930s. Many proof were found, including the surprisingly short proof of Kelly ..."
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Cited by 2 (0 self)
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The SylvesterGallai theorem asserts that any noncollinear point set in the plane determines a line passing through exactly two points in the set. The problem was posed by Sylvester in 1893 and first solved by Gallai in 1930s. Many proof were found, including the surprisingly short proof
A de BruijnErdős theorem and metric spaces
, 2009
"... De Bruijn and Erdős proved that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this theorem in the framework of metric spaces. We provide partial results in this direction. ..."
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Cited by 7 (2 self)
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De Bruijn and Erdős proved that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this theorem in the framework of metric spaces. We provide partial results in this direction.
Incidence Theorems and Their Applications
"... We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: 1. Counting incidences: Given a set (or s ..."
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several sets) of geometric objects (lines, points, etc.), what is the maximum number of incidences (or intersections) that can exist between elements in different sets? We will see several results of this type, such as the SzemerediTrotter theorem, over the reals and over finite fields and discuss
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