In the uniform random intersection graphs model, denoted by Gn;m;, to each vertex v we assign exactly  randomly chosen labels of some label set M of m labels and we connect every pair of vertices that has at least one label in common. In this model, we estimate the independence number (Gn;m;), for the wide, interesting range m = n;  < 1 and  = O(m1=4). We also prove the hamiltonicity of this model by an interesting combinatorial construction. Finally, we give a brief note concerning the independence number of Gn;m;p random intersection graphs, in which each vertex chooses labels with probability p.