>Forced Five Set: A connected subset of five vertices >(not isomorphicc to K5), that cannot be 4-colored.
With the definition as you've stated it, there's no such thing as a forced 5-set.
More generally, there's no such thing as a forced n-set, for any positive integer n.
To be precise, the following is true and easily proved:
If a graph G with n vertices is not complete, then G is (n-1)-colorable.
Thus, your stated definitions of forced 4-set (F4S) and forced 5-set (F5S) are badly flawed, since, if interpreted literally, there are no such sets.
>In the example above, vertices 0 1 7 8 9 constitute an F5S.
No, not the way you've defined it.
The entire graph (on all 10 vertices) is not 4-colorable, but the subgraph determined by vertices 7,8,9,0,1 is definitely 4-colorable. In other words, the subgraph determined by the vertices 7,8,9,0,1 and the edges between them (inherited from the original graph) _is_ 4-colorable.
Let H be the subgraph determined by vertices 7,8,9,0,1. The other vertices (vertices 2,3,4,5,6) and their edges are not part of H. H has only the 5 vertices 7,8,9,0,1 and has exactly 9 edges. In fact, all pairs of distinct vertices of H are adjacent except for the pair 7,1. Thus, if we only want to color the subgraph H (without considering vertices and edges not internal to H), it can be done with 4 colors. For example, using colors A,B,C,D, color the vertices 7,8,9,0,1 as A,B,C,D,A respectively.