  Random $s$-intersection graphs have recently received considerable attention
in a wide range of application areas. In such a graph, each vertex is equipped
with a set of items in some random manner, and any two vertices establish an
undirected edge in between if and only if they have at least $s$ common items.
In particular, in a uniform random $s$-intersection graph, each vertex
independently selects a fixed number of items uniformly at random from a common
item pool, while in a binomial random $s$-intersection graph, each item in some
item pool is independently attached to each vertex with the same probability.
  For binomial/uniform random $s$-intersection graphs, we establish threshold
functions for perfect matching containment, Hamilton cycle containment, and
$k$-robustness, where $k$-robustness is in the sense of Zhang and Sundaram
[IEEE Conf. on Decision & Control '12]. We show that these threshold functions
resemble those of classical Erd\H{o}s-R\'{e}nyi graphs, where each pair of
vertices has an undirected edge independently with the same probability.
