Replace addition by multiplication

Question

On Wikipedia I found the following information:

It is possible to replace addition by multiplication in Folkman's theorem: if the natural numbers are finitely partitioned, there exist arbitrarily large sets S such that all products of nonempty subsets of S belong to a single partition set. Indeed, if one restricts S to consist only of powers of two, then this result follows immediately from the additive version of Folkman's theorem.

Link: https://en.wikipedia.org/wiki/Folkman%27s_theorem

Could someone tell me how to change the multiplication to add?

I would ask for quite exhaustive information.

Is the procedure implementing such calculations very complex? (for only positive integers)

Thanks a lot!

Answer 1

The hint is at the last line in Wiki:

Indeed, if one restricts S to consist only of powers of two, then this result follows immediately from the additive version of Folkman's theorem.

We only need to consider the partition of $\{ 2^n:n \ge 1 \}$. Say it is partitioned into $N_1, \ldots, N_k$. Let $S_i=\{ \ell: 2^{\ell} \in N_i \}$ then $S_1,S_2, \ldots, S_2$ is a partition of $\mathbb{N}$. Thus, according to Folkman's theorem, for any positive integer $m$, there exists set $T_m \subset \mathbb{N}$ so $|T_m|=k$ and every sum of a nonempty subset of $T_m$ belongs to some $S_{i_m}$.

This means if $L_m=\{2^x: x\in T_m\}, |L_m|=m$ then every product of nonempty subset of $L_m$ belongs to $N_{i_m}$.