On the realization of random graphs by graphs of diameters

This work lies on the interface between combinatorial geometry and theory of random graphs. We study the conditions under which a random graph in the Erdos-Renyi model contains subgraphs isomorphic to some graphs of diameters on the plane having their chromatic numbers equal to three. For the appropriate extremal characteristic of a random graph, we succeed in obtaining tight bounds and even asymptotics.