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

Generalised Empty Hexagon Conjecture ★★

Author(s): Wood

Conjecture   For each $ \ell\geq3 $ there is an integer $ f(\ell) $ such that every set of at least $ f(\ell) $ points in the plane contains $ \ell $ collinear points or an empty hexagon.

Keywords: empty hexagon

Nonrepetitive colourings of planar graphs ★★

Author(s): Alon N.; Grytczuk J.; Hałuszczak M.; Riordan O.

Question   Do planar graphs have bounded nonrepetitive chromatic number?

Keywords: nonrepetitive colouring; planar graphs