Combinatorial line

Let $X$ be a finite alphabet. A combinatorial line in $X^n$ is obtained by fixing some coordinates and replacing every coordinate in one nonempty active set by the same variable letter. It therefore contains one word for each letter of $X$.

For every finite alphabet $X$ and every positive integer $k$, there is $n$ such that every $k$-coloring of $X^n$ contains a monochromatic combinatorial line.