Edge labelled graphs and metric spaces

In order to obtain theorems such as:  Every bounded  uncountable separable homogeneous universal metric space is oscillation stable, some quite interesting problems in discrete mathematics had to be addressed.  One of them is the following: Let C be a class of metric spaces and G an edge labelled graph.  Under which conditions on C and G can the labelling of G be extended to a labelling of the complete graph with vertices V(G), so that the metric space on V(G) with distance between two points the label on the edge connecting them, is an element of C.  I will give some indication how a very special case of this problem relates to the theorem above and how this special case can be resolved.