Existence of partitions

Question

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.

Answer 1

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 $\lambda(\mathcal I)> 1/2$.

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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 $$ \bigcup\mathcal I = \mathbb N. $$