Lattices of hereditary properties of nite graphs have been extensively studied. We investigate the lattice L of induced-hereditary
properties of countable graphs. Of interest to us will be some of
the members of L. Much of our focus will be on hom-properties. We analyze their behaviour and consider their link to solving
the long standing Hedetniemi Conjecture. We then discuss universal
graphs and construct a universal graph for hom-properties.
We then use these universal graphs to prove a theorem by Szekeres and
Wilf. Lastly we off er a new proof of a theorem by Du ffus, Sands and