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Por, Attila


Geometry

Geometric Hales-Jewett Theorem ★★

Author(s): Por; Wood

Conjecture   For all integers $ k\geq1 $ and $ \ell\geq3 $, there is an integer $ f(k,\ell) $ such that for every set $ P $ of at least $ f(k,\ell) $ points in the plane, if each point in $ P $ is assigned one of $ k $ colours, then:
    \item $ P $ contains $ \ell $ collinear points, or \item $ P $ contains a monochromatic line (that is, a maximal set of collinear points receiving the same colour)

Keywords: Hales-Jewett Theorem; ramsey theory

Posted by David Wood
updated March 26th, 2014
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Geometry

Big Line or Big Clique in Planar Point Sets ★★

Author(s): Kara; Por; Wood

Let $ S $ be a set of points in the plane. Two points $ v $ and $ w $ in $ S $ are visible with respect to $ S $ if the line segment between $ v $ and $ w $ contains no other point in $ S $.

Conjecture   For all integers $ k,\ell\geq2 $ there is an integer $ n $ such that every set of at least $ n $ points in the plane contains at least $ \ell $ collinear points or $ k $ pairwise visible points.

Keywords: Discrete Geometry; Geometric Ramsey Theory

Posted by David Wood
updated September 25th, 2012
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