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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 ..."
Abstract

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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 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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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 using Euclidean distances, and the one by Melchoir using Euler’s formula. We survey the history of the theorem and related problems, including various proofs of the classical SylvesterGallai theorem, the lower bound on the number of Gallai lines, the deBruijnErdős theorem, the Scott’s conjecture and Ungar’s theorem, the Dirac conjecture, the magic configuration conjecture, the question on the number of Gallai points, and the colored version of the problem. We then present the recent work on the generalization of these problems in arbitrary metric space and hypergraphs. In particular, we present the SylvesterChvátal theorem and the problems related to the de BruijnErdős theorem. Another problem we study in this dissertation is the visibility of points and segments in the plane. We color the end points of each segments red and blue, and study, in particular, the visibility relations between the red points and the blue ones. We