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Every <finite coloring> of the <positive integer>[positive integers] contains a monochromatic <Hilbert cube> of every prescribed finite dimension.

Hilbert cube theorem

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 integer>
$$
\Phi(w)=1+\sum_{j=1}^N 2^{j-1}w_j.
$$
A monochromatic $n$-parameter set has disjoint active coordinate sets $I_1,\ldots,I_n$ and fixed letters outside their union. Put
$$
a=1+\sum_{j\notin I_1\cup\cdots\cup I_n}2^{j-1}w_j,
\qquad
x_s=\sum_{j\in I_s}2^{j-1}.
$$
Every $x_s$ is positive. As the $n$ independent variable letters range over zero and one, their images under $\Phi$ are precisely
$$
\left\{a+\sum_{s\in J}x_s:J\subseteq[n]\right\}.
$$
All these integers have one color, so they form the required monochromatic <Hilbert cube>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 130 / 1 / ii / Solution

A Hilbert $n$-cube in the <positive integer>[positive integers] is a set
$$
\left\{a+\sum_{i\in I}x_i:I\subseteq[n]\right\},
$$
where $a,x_1,\ldots,x_n$ are positive integers.

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