Optimal and Near-Optimal Constructions for Bootstrap Percolation in Hypercubes

TL;DR

This paper determines the minimal initial infected set size for 4-neighbour bootstrap percolation in hypercubes, providing exact values for infinitely many dimensions and bounds for all dimensions.

Contribution

It establishes the exact value of the minimal infected set size for 4-neighbour bootstrap percolation in hypercubes for infinitely many dimensions, extending previous bounds.

Findings

Exact formula for m(Q_d;4) for infinitely many d

Upper bounds on m(Q_d;4) differing by O(d) from lower bounds

Key constructions aided by AlphaEvolve

Abstract

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.