  A $d$-dimensional grid is a set of the form $R = [a_1] \times ... \times
[a_d]$. A $d$-dimensional box is a set of the form $\{b_1,c_1\} \times ...
\times \{b_d,c_d\}$. When a grid is $c$-colored, must it admit a monochromatic
box? If so, we say that $R$ is $c$-guaranteed. This question is a relaxation of
one attack on bounding the van der Waerden numbers, and also arises as a
natural hypergraph Ramsey problem (viz. the Ramsey numbers of hyperoctahedra).
We give conditions on the $a_i$ for $R$ to be $c$-guaranteed that are
asymptotically tight, and analyze the set of minimally $c$-guaranteed grids.
