Synchronization and separation in the Johnson schemes

Abstract

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 synchronizing and 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 $\{1,\dots ,n\}$ 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 $n\ge 10$ (it fails to be synchronizing for $n=9$).