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 $n=9$).

中文翻译：

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Johnson 方案中的同步和分离

最近，Peter Keevash 通过证明只要满足参数的必要可分条件并且 $n$ 就 $k$ 和 $ 而言足够大，$S(t,k,n)$ 就存在，从而渐近地解决了 Steiner 系统的存在问题t$。本文的目的是做出一个猜想，如果该猜想为真，将是 Keevash 定理的重要扩展，并为该猜想提供一些理论和计算证据。我们根据关联方案的同步和分离的概念（我们在此定义）来表述该猜想。这些定义基于从有限自动机中的同步理论发展而来的置换群的定义。在这个理论中，定义了位于原始和 $2$-homogeneous 之间的两类置换组（称为 \emph{synchronizing} 和 \emph{separating}）。一个很大的悬而未决的问题是由 $S_n$ 在 $\{1,\ldots,n\}$ 的 $k$-子集上引起的置换群如何适合这个层次结构；我们的猜想会给出一个解决这个问题的解决方案，以 $k$ 表示 $n$ 大。我们在 $k=4$ 的情况下证明了这个猜想：我们的结果断言，作用于 $4$-sets 的 $S_n$ 正在分离 $n\ge10$（它无法同步 $n=9$）。