  Recently Peter Keevash solved asymptotically the existence question for
Steiner systems by showing that $S(t,k,n)$ exists whenever the necessary
divisibility conditions on the parameters are satisfied and $n$ is sufficiently
large in terms of $k$ and $t$. The purpose of this paper is to make a
conjecture which if true would be a significant extension of Keevash's theorem,
and to give some theoretical and computational evidence for the conjecture.
  We phrase the conjecture in terms of the notions (which we define here) of
synchronization and separation for association schemes. These definitions are
based on those for permutation groups which grow out of the theory of
synchronization in finite automata. In this theory, two classes of permutation
groups (called \emph{synchronizing} and \emph{separating}) lying between
primitive and $2$-homogeneous are defined. A big open question is how the
permutation group induced by $S_n$ on $k$-subsets of $\{1,\ldots,n\}$ fits in
this hierarchy; our conjecture would give a solution to this problem for $n$
large in terms of $k$. We prove the conjecture in the case $k=4$: our result
asserts that $S_n$ acting on $4$-sets is separating for $n\ge10$ (it fails to
be synchronizing for $n=9$).
