  The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that
whenever $[n]$ is $r$--colored there exists a monochromatic arithmetic
progression of length $k$. Similarly, the Hilbert cube number $h(k,r)$ denotes
the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a
monochromatic affine $k$--cube, that is, a set of the form$$\left\{x_0 +
\sum_{b \in B} b : B \subseteq A\right\}$$ for some $|A|=k$ and $x_0 \in
\mathbb{Z}$.
  We show the following relation between the Hilbert cube number and the Van
der Waerden number. Let $k \geq 3$ be an integer. Then for every $\epsilon >0$,
there is a $c > 0$ such that $$h(k,4) \ge \min\{W(\lfloor c k^2\rfloor, 2),
2^{k^{2.5-\epsilon}}\}.$$ Thus we improve upon state of the art lower bounds
for $h(k,4)$ conditional on $W(k,2)$ being significantly larger than $2^k$. In
the other direction, this shows that the if the Hilbert cube number is close
its state of the art lower bounds, then $W(k,2)$ is at most doubly exponential
in $k$.
  We also show the optimal result that for any Sidon set $A \subset
\mathbb{Z}$, one has $$\left|\left\{\sum_{b \in B} b : B \subseteq
A\right\}\right| = \Omega( |A|^3) .$$
