Bootstrap percolation, introduced by Chalupa, Leath, and Reich in their 1979 work [4], serves as a simplified model for ferromagnetic dynamics. Since then, it has been applied in numerous fields of physics and has become a significant topic of interest in mathematics. The process starts with an initial set of "infected" vertices in a graph G, where at each step, any vertex with at least r infected neighbors also becomes infected. A key problem in this framework is determining the minimum size of an initial set that results in the entire graph becoming infected, known as the percolating set. This minimum is denoted by m(G, r).

We introduce PatternBoost, a flexible method for finding interesting constructions in mathematics. Our algorithm alternates between two phases. In the first ``local'' phase, a classical search algorithm is used to produce many desirable constructions. In the second ``global'' phase, a transformer neural network is trained on the best such constructions. Samples from the trained transformer are then used as seeds for the first phase, and the process is repeated. We give a detailed introduction to this technique, and discuss the results of its application to several problems in extremal combinatorics. The performance of PatternBoost varies across different problems, but there are many situations where its performance is quite impressive. Using our technique, we find the best known solutions to several long-standing problems, including the construction of a counterexample to a conjecture that had remained open for 30 years.

This work studies a central extremal graph theory problem inspired by a 1975 conjecture of Erd\H{o}s, which aims to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this problem as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum -- jump-starting the search for larger graphs using good graphs found at smaller sizes -- we improve the state-of-the-art lower bounds for several sizes. We also propose a flexible graph-generation environment and a permutation-invariant network architecture for learning to search in the space of graphs.