The coupling method for inhomogeneous random intersection graphs

We present new results concerning threshold functions for a wide family of random intersection graphs. To this end we apply the coupling method used for establishing threshold functions for homogeneous random intersection graphs introduced by Karo'nski, Scheinerman, and Singer–Cohen. In the case of inhomogeneous random intersection graphs the method has to be considerably modified and extended. By means of the altered method we are able to establish threshold functions for a general random intersection graph for such properties as $k$-connectivity, matching containment or hamiltonicity. Moreover using the new approach we manage to sharpen the best known results concerning homogeneous random intersection graph.