Hedetniemi's Conjecture and Dense Boolean Lattices

The category D of finite directed graphs is cartesian closed, hence it has a product and exponential objects. For a fixed K, let K be the class of all directed graphs of the formK, preordered by the existence of homomorphisms, and quotiented by homomorphic equivalence. It has loong been known that K, is always boolean lattice. In this paper we prove that for any complete graph Kn with n ≥ 3, K n is dense, hence up to isomorphism it is the unique countable dense boolean lattice. In graph theory, the structure of K n is connected to the conjecture of Hedetniemi on the chromatic number of a categorical product of graphs.