# discrete mathematics – Split all subsets of \$mathcal{P}(mathbb{N})\$ into 2 groups \$A dotcup B =mathcal{P}mathbb(N)\$ s.t. no 2 neighboring sets are in the same group

So I’m trying to come up with a proof of the above action being possible or not. I feel like I could use the compactness theorem on this one but I’m not quite sure how to split up the finite subsets of $$mathbb{n}$$ to make the condition true.

I’d really appreciate some help on this :] Thanks so much!