## Geometric Hales-Jewett Theorem ★★

**Conjecture**For all integers and , there is an integer such that for every set of at least points in the plane, if each point in is assigned one of colours, then:

- \item contains collinear points, or \item 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 there is an integer such that every set of at least points in the plane contains 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