the layers of the cube are just the number of elements in the set, for the cube on 4 elements 1,2,3,4, we have the coloring where every set containing 1 is blue

the layers of the cube are just the number of elements in the set, for the cube on 4 elements 1,2,3,4, we have the coloring where every set containing 1 is blue. certainly there is a blue one element set {1} so by your intuition, every three element set must be blue, but {2,3,4} does not contain 1, so it is white.