Rainbow considerations around the Hales-Jewett theorem

For positive integers $t$ and $n$ let $C_t^n$ be the $n$-cube over $t$ elements, that is, the set of ordered $n$-tuples over the alphabet $\{0,\dots ,t-1\}$. We address the question of whether a balanced finite coloring of $C_t^n$ guarantees the presence of a rainbow geometric or combinatorial line. For every even $t\ge 4$ and every $n$, we provide a ${\left(\frac{t}{2}\right)}^n$–coloring of $C_t^n$ such that all color classes have the same size, and without rainbow combinatorial or geometric lines.