A note on Hamiltonicity of uniform random intersection graphs

We consider a collection of n independent random subsets of [m] = {1, 2, . . . , m} that are uniformly distributed in the class of subsets of size d, and call any two subsets adjacent whenever they intersect. This adjacency relation defines a graph called the uniform random intersection graph and denoted by Gn,m,d. We fix d = 2, 3, . . . and study when, as n,m → ∞, the graph Gn,m,d contains a Hamilton cycle (the event denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {G_{n,m,d}} \in \mathcal{H} $\end{document}). We show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {\mathbf{P}}\left( {{G_{n,m,d}} \in \mathcal{H}} \right) = o(1) $\end{document} for d2nm−1− lnm − 2 ln lnm → −∞ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {\mathbf{P}}\left( {{G_{n,m,d}} \in \mathcal{H}} \right) = 1 - o(1) $\end{document} for 2nm−1− lnm − ln lnm → +∞.