The $r$-neighbour bootstrap process on a graph $G$ begins with a set of infected vertices; subsequently, healthy vertices become infected once they have at least $r$ infected neighbours. The central extremal problem in bootstrap percolation is to determine the minimum cardinality of an initial infected set that eventually spreads to all vertices of $G$, denoted $m(G;r)$. Morrison and Noel established a general lower bound on $m(Q_d;r)$, where $Q_d$ is the $d$-dimensional hypercube, and asked whether it is tight whenever $d$ is sufficiently large with respect to $r$. This question was answered affirmatively for $r\leq 3$. In this paper, we show that $m(Q_d;4)=\frac{d(d^2+3d+14)}{24}+1$, matching the bound in of Morrison and Noel, for infinitely many $d$. We also obtain, for general $d$, an upper bound on $m(Q_d;4)$ that differs from the Morrison--Noel lower bound by an additive $O(d)$ term. Several key constructions in this paper were obtained with the assistance of AlphaEvolve.

翻译：在给定图$G$上，$r$-近邻自举过程从一组初始感染顶点开始；此后，健康顶点一旦拥有至少$r$个感染近邻即被感染。自举渗滤的核心极值问题在于确定最终能感染$G$全部顶点的最小初始感染集基数，记为$m(G;r)$。Morrison与Noel建立了$m(Q_d;r)$的一般下界（其中$Q_d$为$d$维超立方体），并询问当$d$相对于$r$足够大时该下界是否紧确。对于$r\leq 3$的情况，该问题已得到肯定解答。本文中，我们证明在无穷多个$d$值下$m(Q_d;4)=\frac{d(d^2+3d+14)}{24}+1$，符合Morrison与Noel的下界。此外，对于一般$d$，我们给出$m(Q_d;4)$的上界，该上界与Morrison-Noel下界仅相差一个$O(d)$的加性项。本文的若干关键构造由AlphaEvolve辅助完成。