The minimal degree of a permutation group G is the minimum number of points not fixed by non-identity elements of G. Lower bounds on the minimal degree have strong structural consequences on G. Babai conjectured that if a primitive coherent configuration with n vertices is not a Cameron scheme, then its automorphism group has minimal degree $\ge cn$ for some constant $c>0$. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3.

In this paper, we extend Babai's result to primitive coherent configurations of rank 4, confirming the conjecture in this special case. The proofs combine structural and spectral methods.

Recently (March 2022) Sean Eberhard published a class of counterexamples of rank 28 to Babai's conjecture and suggested to replace “Cameron schemes” in the conjecture with a more general class he calls “Cameron sandwiches”. Naturally, our result also confirms the rank 4 case of Eberhard's version of the conjecture.

置换群G的最小度是G的非恒等元素未固定的点的最小数量。最小程度的下界对G有很强的结构影响。巴拜猜想，如果一个有n个顶点的本原相干构型不是卡梅伦方案，那么它的自同构群具有最小度$\ge Cn$对于一些常数$C>0$。2014年，Babai证明了强正则图自同构群最小度所需的下界，从而证实了3阶本原相干构型的猜想。

在本文中，我们将Babai的结果扩展到4阶的原始相干构型，证实了这种特殊情况下的猜想。证明结合了结构方法和谱方法。

最近（2022 年 3 月）Sean Eberhard 发表了 Babai 猜想的 28 级反例，并建议用一个更通用的类（他称之为“Cameron 三明治”）取代猜想中的“Cameron 方案”。当然，我们的结果也证实了艾伯哈德猜想版本的排名 4 的情况。