Good morning everybody.

I would like to know if anybody is aware of nontrivial results of the following form : *if a family $\mathcal I$ of subsets of $\mathbb N$ satisfies such and such assumption, then one may conclude that $\mathbb N$ can be partitioned into finitely many sets from $\mathcal I$.*

This is of course quite vague, but the general idea should be clear.

Thanks in advance.