Hales-Jewett Number $HJ(2,r)$ I'm trying to find the exact value for the Hales-Jewett number $HJ(2,r)$, where $HJ(k,r)$ is defined as the smallest $n$ so that any coloring of the elements of $[k]^n$ by $r$ colors has a monochromatic combinatorial line.
It seems like a simple (maybe even trivial) problem, but I'm not sure how to proceed since I'm still trying to wrap my head around combinatorial lines.

Consider the points in $[2]^r$ of the form:
$(1,1,1,\ldots, 1),$
$(2,1,1,\ldots, 1)$,
$(2,2,1,\ldots, 1),$
$\qquad\ddots$
$(2,2,2,\ldots, 2)$.
Any pair of these form a combinatorial line.  As there $r+1$ of these points, in any $r$-colouring, some pair must have the same colour.
Conversely, if $k<r$, then we may $r$-colour $[2]^k$ by number of co-ordinates equal to $2$.  Then any monochromatic pair of points will not form a combinatorial line, as moving from one point to the other, there will exist a co-ordinate where a $1$ changes to a $2$, as well as a co-ordinate where a $2$ changes to a $1$.
Thus we have $HJ(2,r)=r$.
