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. 



Existence of partitions
Let $\lambda$ denote the uniform measure on the powerset of $\mathbb N$ (also known as the Lebesgue measure and the fair-coin measure).
Let $\tau$ be the product topology of the discrete topology on $\{0,1\}$.


In order to be able to conclude that $\mathbb N$ can be partitioned into finitely many sets from $\mathcal I$ (in fact, *two*),


* it is sufficient that $\mathcal I $ be comeager with respect to $\tau$;
* it is sufficient to have $\lambda(\mathcal I)> 1/2$.




---


These observations are sharp, as the example $\mathcal I:=\{X: 17\in X\}$ shows: it is not sufficient that $\mathcal I $ be nonmeager, nor that $\mathcal I $ be of Lebesgue measure $\ge 1/2$, because it is necessary that
$$
\bigcup\mathcal I = \mathbb N.
$$


