OPEN
Is there a function $f(n)$ and a $k$ such that in any $k$-colouring of the integers there exist, for infinitely many $n$, a sequence $a_1<\cdots <a_n<f(n)$ such that the set
\[\left\{ \sum_{i\in S}a_i : S\subseteq [n]\right\}\]
does not contain all colours?
Erdős initially asked whether this is possible with the set being monochromatic, but Galvin showed that this is not always possible, considering the two colouring where, writing $n=2^km$ with $m$ odd, we colour $n$ red if $m\geq F(k)$ and blue if $m<F(k)$ (for some sufficiently quickly growing $F$).
This is open even in the case of $\aleph_0$-many colours.
This is asking about a variant of Hindman's theorem (see [532]).
