# Geometric Hales-Jewett Theorem (Solved)

**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)

This conjecture is trivially true for with , and for and . The Motzkin-Rabin Theorem [PS] says that the conjecture is true for with . The conjecture is related to the Hales-Jewett Theorem, which states that for sufficiently large , every -colouring of the -dimensional grid contains a monochromatic "combinatorial"' line of points. Say the *Geometric Hales-Jewett Theorem* is the statement obtained by replacing "combinatorial line" by "geometric line" (that is, a set of collinear points). This theorem is discussed in [P]. The conjecture is a generalisation of the Geometric Hales-Jewett Theorem.

This conjecture was disproved by Vytautas Gruslys [G].

## Bibliography

[P] D.H.J. Polymath. Density Hales-Jewett and Moser numbers, arXiv:1002.0374, 2010.

*[PW] Attila Pór and David R. Wood. On visibility and blockers, J. Computational Geometry 1:29-40, 2010.

[PS] Lou M. Pretorius and Konrad J. Swanepoel. An Algorithmic Proof of the Motzkin-Rabin Theorem on Monochrome Lines, The American Mathematical Monthly 111.3:245-251, 2004.

[G] Vytautas Gruslys. A counterexample to a geometric Hales-Jewett type conjecture. J. Comput. Geom. 5.1:245-249, 2014.

* indicates original appearance(s) of problem.