Rainbow considerations around the Hales-Jewett theorem

Abstract: For positive integers $t$ and $n$ let $C_t<sup>n$ 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 $C_t<sup>n$ guarantees the presence of a rainbow geometric or combinatorial line. For every even $t\geq 4$ and every $n$, we provide a $\left(\frac{t}{2}\right)<sup>n$--coloring of $C_t<sup>n$ such that all color classes have the same size, and without rainbow combinatorial or geometric lines.