Optimal and Near-Optimal Constructions for Bootstrap Percolation in Hypercubes

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.

• The paper presents explicit and recursive constructions for r=4 bootstrap percolation in hypercubes, achieving optimal or near-optimal percolating set sizes.
• It develops a novel meta-bootstrap framework using nested percolating sets and AI-assisted methods like AlphaEvolve to verify small-dimensional cases.
• The results close a significant gap by matching the Morrison–Noel lower bound for infinitely many dimensions and provide a framework applicable to broader percolation models.

Optimal and Near-Optimal Bootstrap Percolating Set Constructions in Hypercubes

Introduction and Problem Context

This work addresses a central extremal problem in bootstrap percolation on the $d$-dimensional hypercube $Q_d$: determining the minimum cardinality $m(Q_d; r)$ of an initial vertex set that percolates via the $r$-neighbour process. Specifically, the paper focuses on the $r=4$ case, where a healthy vertex becomes infected if it has at least 4 infected neighbors, furthering foundational results established by Morrison and Noel [MorrisonNoel18]. Their general lower bound on $m(Q_d; r)$ remains a benchmark, with tightness verified for $r \leq 3$. The principal contributions here are explicit constructions for $r=4$ achieving or nearly achieving this extremal bound for infinitely many $d$, thus resolving a significant open case in the theory.

Theoretical Advances and Main Results

The paper establishes the following central result for infinitely many $d$: $Q_d$0 whenever $Q_d$1, $Q_d$2, or $Q_d$3. This exactly matches the Morrison–Noel lower bound [MorrisonNoel18]. For general $Q_d$4, the construction yields an upper bound on $Q_d$5 within an additive $Q_d$6 term of the lower bound, formalized as: $Q_d$7 This provides asymptotically tight estimates up to an additive constant for arbitrary hypercube dimensions.

Key technical advances include the development of recursive, block-based constructions and the introduction of a meta-bootstrap process framework that employs percolating labelings to organize percolation propagation within subcubes. All constructions guarantee nesting of smaller $Q_d$8-neighbour percolating sets ($Q_d$9), ensuring compatibility for the recursive build-up required in Lemma 3.1. The explicit small-dimension constructions (for $m(Q_d; r)$0) were obtained using AlphaEvolve [Novikov25], a generative agent-based system designed for computational mathematical discovery.

Methodology and Construction Principles

The construction approach generalizes and refines prior techniques [BaloghBollobas06, MorrisonNoel18, BaloghBollobasMorrisRiordan12]. Several critical methodological components are as follows:

1. Meta-Bootstrap Process: The percolation process is encoded as a meta-propagation over a quotient graph formed by partitioning $m(Q_d; r)$1 into subcubes $m(Q_d; r)$2. These meta-level dynamics are controlled via percolating labelings which determine the nesting, subdivision, and activation of percolating sets within each subcube.
2. Nested Percolating Sets: For each dimension $m(Q_d; r)$3, percolating sets $m(Q_d; r)$4 for $m(Q_d; r)$5 are explicitly constructed, enabling recursive application of Lemma 3.1. These constructions are provided with optimal or near-optimal cardinality and demonstrated to propagate infection in minimal rounds.
3. Explicit Small-dimension Seeds: For $m(Q_d; r)$6 with small $m(Q_d; r)$7, explicit sets $m(Q_d; r)$8 are listed and their percolating properties are verified via computational enumeration and accompanying code. These seeds anchor the recursive extension to higher dimensions.
4. Inductive/Recursive Step: The construction for large $m(Q_d; r)$9 is built as a combination of subcube percolating sets, orchestrated via percolating labelings (using newly defined meta-bootstrap processes), ensuring a cascade of infection which covers the entire hypercube with minimal initial "budget".

Comparison to Prior Upper and Lower Bounds

Previous upper-bound constructions (e.g., the approximate Steiner system strategy [BaloghBollobas06, BaloghBollobasMorrisRiordan12]) yield upper bounds matching the Morrison–Noel lower bound in the leading order term but exhibit gaps in the second-order terms for $r$0. Bérczi and Wagner [BercziWagner24] further improved these using AI-generated percolating sets, especially for small $r$1, but the present work surpasses previous constructions by matching the lower bound exactly for infinitely many $r$2 without assuming the existence of exact Steiner systems.

The core recursive construction provides strong numerical guarantees (exact for many $r$3, within $r$4 otherwise) and bypasses the limitations imposed by combinatorial design dependencies in prior approaches.

Computational and Algorithmic Components

A substantial part of the discovery and verification process utilized AlphaEvolve [Novikov25], which autonomously explored the search space for percolating sets and labelings, subject to formal constraints and recursive composition requirements. The authors then verified all percolation properties of the produced objects independently through customized code, whose source and execution records are included. This systematic integration of generative computational agents marks an innovative advance in extremal combinatorics methodology and is notable for the scale of the enumeration and verification exercises required.

Implications and Future Directions

The results provide a resolution for the $r$5 case of the minimum percolating set problem in hypercubes for an infinite sequence of $r$6, solidifying the tightness of the Morrison–Noel lower bound in these cases. The theoretical implications are significant, demonstrating that recursive meta-bootstrap constructions, supported by AI-based search for seed configurations, are sufficient to close the gap between lower and upper bounds for $r$7 beyond what previous combinatorial techniques allowed.

Practically, the methods suggest a general framework for constructing minimal percolating sets for arbitrary locality-constrained percolation models on high-dimensional graphs, with direct applications to information propagation, coding theory, and distributed network activation protocols. The integration of generative AI systems like AlphaEvolve for discovering extremal combinatorial structures points toward broader applications in computer-assisted combinatorics and extremal graph theory.

On a theoretical level, the primary conjecture remains that for each fixed $r$8, the Morrison–Noel lower bound is eventually tight for all sufficiently large $r$9, barring isolated exceptions (notably, $r=4$0 for $r=4$1). Realizing this in full generality for higher $r=4$2 likely requires new conceptual and algorithmic breakthroughs, as the complexity and verification demands outstrip brute-force and current recursive approaches for large $r=4$3.

Conclusion

This paper delivers both exact and near-optimal constructions for bootstrap percolation thresholds in hypercubes for $r=4$4, resolving a key case in extremal percolation theory. These results significantly extend prior approaches and leverage hybrid algebraic, combinatorial, and computational techniques, including the use of generative coding agents for explicit configuration search. The frameworks and methods introduced here will likely inform future research on percolation, extremal design problems, and algorithmic combinatorics, especially as AI-assisted discovery becomes more prevalent in mathematical exploration.

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References

• N. Morrison and J. A. Noel, "Extremal bounds for bootstrap percolation in the hypercube" [MorrisonNoel18]
• A. Novikov et al., "AlphaEvolve: A coding agent for scientific and algorithmic discovery" [Novikov25]
• G. Bérczi and A. Zs. Wagner, "A Note on Small Percolating Sets on Hypercubes via Generative AI" [BercziWagner24]
• J. Balogh and B. Bollobás, "Bootstrap percolation on the hypercube" [BaloghBollobas06]
• J. Balogh, B. Bollobás, R. Morris, and O. Riordan, "Linear algebra and bootstrap percolation" [BaloghBollobasMorrisRiordan12]