The Sylvester-Gallai Theorem, the Monochrome Line Theorem and Generalizations Report for a Seminar on the Sylvester-Gallai Theorem
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منابع مشابه
Extremal Problems Related to the Sylvester–Gallai Theorem
We discuss certain extremal problems in combinatorial geometry, including Sylvester’s problem and its generalizations.
متن کاملThe Sylvester-Chvatal Theorem
The Sylvester-Gallai theorem asserts that every finite set S of points in two-dimensional 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. Chvátal extended the notion of lines to arbitrary metric spaces and made a conjecture that generalizes the Sylvester-Gallai theorem. In the present ...
متن کاملA new proof of the Sylvester-Gallai theorem
In 1893 J. J. Sylvester [8] posed the following celebrated problem: Given a finite collection of points in the affine plane, not all lying on a line, show that there exists a line which passes through precisely two of the points. Sylvester’s problem was reposed by Erdős in 1944 [4] and later that year a proof was given by Gallai [6]. Since then, many proofs of the Sylvester-Gallai Theorem have ...
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 کاملA Reverse Analysis of the Sylvester-Gallai Theorem
Reverse analyses of three proofs of the Sylvester-Gallai theorem lead to three different and incompatible axiom systems. In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.
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تاریخ انتشار 2012