OPEN

Let $c(n)$ be minimal such that if $k\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good bounds for $c(n)$ - in particular, is it true that $c(n) \gg n^n$?

A problem first investigated by Hadwiger, who proved the lower bound
\[c(n) \geq 2^n+2^{n-1}.\]
It is easy to see that $c(2)=6$. Meier conjectured $c(3)=48$. Burgess and Erdős [Er74b] proved
\[c(n) \ll n^{n+1}.\]
Erdős wrote 'I am certain that if $n+1$ is a prime then $c(n)>n^n$.'.