  Ramsey theory looks for regularities in large objects. Model theory studies
algebraic structures as models of theories. The structural Ramsey theory
combines these two fields and is concerned with Ramsey-type questions about
certain model-theoretic structures. In 2005, Ne\v{s}et\v{r}il initiated a
systematic study of the so-called Ramsey classes of finite structures. This
thesis is a contribution to the programme; we find Ramsey expansions of the
primitive 3-constrained classes from Cherlin's catalogue of metrically
homogeneous graphs. A key ingradient is an explicit combinatorial algorithm to
fill-in the missing distances in edge-labelled graphs to obtain structures from
Cherlin's classes. This algorithm also implies the extension property for
partial automorphisms (EPPA), another combinatorial property of classes of
finite structures.
