# Synchronization and separation in the Johnson schemes

## TL;DR

This paper explores the concepts of synchronization and separation in association schemes, proposing a conjecture that extends Keevash's theorem on Steiner systems, supported by theoretical and computational evidence.

## Contribution

It introduces the notions of synchronization and separation in association schemes and conjectures their relation to permutation groups, providing proof for the case k=4.

## Key findings

- Proves that S_n acting on 4-sets is separating for n≥10.
- Shows S_n on 4-sets is not synchronizing for n=9.
- Provides computational evidence supporting the conjecture.

## 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 \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 $n=9$).