![](/files/happy5.png)
![$ k\geq1 $](/files/tex/636eac0b23e9e5ba1f1a1fc5e22e2d2009ff1533.png)
![$ \ell\geq3 $](/files/tex/06509000d5e4893e990a6bf4e2deca4af4e82a6c.png)
![$ f(k,\ell) $](/files/tex/b9ec548be74b7eb11cfc1d7f1b688bc508002543.png)
![$ P $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
![$ f(k,\ell) $](/files/tex/b9ec548be74b7eb11cfc1d7f1b688bc508002543.png)
![$ P $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
- \item
![$ P $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
![$ P $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
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.