Suppose we have $2^n$ elements in a set. We have $cn^\beta$ random subsets of cardinality $\frac{2^n}{c}$ elements each where $c,\beta>1$ holds.
Fix a random subset of $n^\alpha$ elements $A$ where $\alpha>1$.
(1) What is the probability that there is at least one subset which will pick every element in $A$?
(2) How many subsets of size $n^\alpha$ distinct elements are not covered by any of the $cn^\beta$ subsets?
(3) What is the minimum number of subsets to cover every subset of size $n^\alpha$ elements?