If we have a set of elements and we can split them into different subsets. We can color the elements either red or blue. Check if there is a way to color the elements such that there is no subset that contains one color.
I want to prove this problem is NP-Complete using boolean SAT (i.e. I need to reduce SAT to this problem).
Any variation of the SAT problem is fine. E.g. 3 SAT.
Hint. For each-variable in a CNF-SAT problem have two elements in your set, representing $x$ and $\neg x$. Then emit the 2-element subset $\{x,\neg x\}$ to force them to have opposite colors.
Also have a special global element $0$. It can end up being either red or blue, but whatever it ends up being will encode "false" (and the other color will then be "true").
Can you now see how to encode a single clause as a subset-that-must-not-be-monochrome?