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combinatorial geometry
The Double Cap Conjecture ★★
Author(s): Kalai
Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of
containing no pair of orthogonal vectors is attained by two open caps of geodesic radius
around the north and south poles.
![$ \mathbb{R}^n $](/files/tex/2010c953180b3521ec2f66d10e1f40ec71d44574.png)
![$ \pi/4 $](/files/tex/01608ea3b80f85b77096d16610a43e184782386c.png)
Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere
Point sets with no empty pentagon ★
Author(s): Wood
Problem Classify the point sets with no empty pentagon.
Keywords: combinatorial geometry; visibility graph
Erdös-Szekeres conjecture ★★★
Conjecture Every set of
points in the plane in general position contains a subset of
points which form a convex
-gon.
![$ 2^{n-2} + 1 $](/files/tex/78669eb6d8b95f5ad57efe25b8ed4f27b42cb6f6.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
Keywords: combinatorial geometry; Convex Polygons; ramsey theory
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