Summary: We study properties of the uniform random intersection graph model $G(n,m,d)$. We find asymptotic estimates on the diameter of the largest connected component of the graph near the phase transition and connectivity thresholds. Moreover we manage to prove an asymptotically tight bound for the connectivity and phase transition thresholds for all possible ranges of $d$, which has not been obtained before. The main motivation of our research is the usage of the random intersection graph model in the studies of wireless sensor networks.