  For a linear block code C, its stopping redundancy is defined as the smallest
number of check nodes in a Tanner graph for C, such that there exist no
stopping sets of size smaller than the minimum distance of C. Schwartz and
Vardy conjectured that the stopping redundancy of an MDS code should only
depend on its length and minimum distance.
  We define the (n,t)-single-exclusion number, S(n,t) as the smallest number of
t-subsets of an n-set, such that for each i-subset of the n-set, i=1,...,t+1,
there exists a t-subset that contains all but one element of the i-subset. New
upper bounds on the single-exclusion number are obtained via probabilistic
methods, recurrent inequalities, as well as explicit constructions. The new
bounds are used to better understand the stopping redundancy of MDS codes. In
particular, it is shown that for [n,k=n-d+1,d] MDS codes, as n goes to
infinity, the stopping redundancy is asymptotic to S(n,d-2), if d=o(\sqrt{n}),
or if k=o(\sqrt{n}) and k goes to infinity, thus giving partial confirmation of
the Schwartz-Vardy conjecture in the asymptotic sense.
