 Solution

Let the given coloring use $k$ colors, and choose $N$ from the Extended Hales-Jewett theorem for alphabet $\{0,1\}$ and dimension $n$. Color a word $w\in \{0,1{\}}^N$ by the color of the positive integerA monochromatic $n$-parameter set has disjoint active coordinate sets $I_1,\dots ,I_n$ and fixed letters outside their union. PutEvery $x_s$ is positive. As the $n$ independent variable letters range over zero and one, their images under $\Phi$ are preciselyAll these integers have one color, so they form the required monochromatic Hilbert cube.