to be interpreted as the sets of vertices colored with the 3 colors, and E(v, w) is to be interpreted as saying the vertices v, w share an edge. The formula then states that no two adjacent vertices have the same color.

Note that there is no need to explicitly enforce each vertex to have exactly 1 color, just at least 1 color. Since, if we allow more than 1 color per vertex, and no two adjacent vertices have the same color, then we certainly can have each vertex to have exactly 1 color, and still no two adjacent vertices have the same