Abstract: In the $n$-dimensional grid $[t]^n = \{1, 2, \dots, t\}^n$, a combinatorial line is an injective function $\ell : [t] \to [t]^n$ such that for each coordinate $1 \le i \le n$, $\ell_i$ is either constant or the identity function on $[t]$. An example of such a line in $[4]^5$ is the function $\ell(x) = (3, x, 1, x, 4)$ whose image is the set of four points (3,1,1,1,4), (3,2,1,2,4), (3, 3,1,3,4), (3,4,1,4,4). A classic result in Ramsey theory, the Hales-Jewett theorem, asserts that for all values of the parameters $t$ and $r$, there is a sufficiently large $n = HJ(t,r)$ such that any $r$-coloring of $[t]^n$ will contain a combinatorial line whose points are monochromatic. Apart from $HJ(2,r)$ for variable $r$ and $HJ(3,2)$ which have all been determined exactly, good bounds on values of $HJ(t,r)$ are generally not known. Even for the small case $HJ(4,2)$, the best result known is a general result due to Shelah, which gives an upper bound between $2 \uparrow \uparrow 7$ and $2 \uparrow \uparrow 8$: quite far from the best known lower bound of 12. We show that the upper bound can be improved to $10^{11}$.