The r-neighbour bootstrap percolation process on a graph G starts with an initial set A 0 of "infected" vertices and, at each step of the process, a healthy vertex becomes infected if it has at least r infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of G eventually becomes infected, then we say that A 0 percolates.We prove a conjecture of Balogh and Bollobás which says that, for fixed r and d → ∞, every percolating set in the d-dimensional hypercube has cardinality at least 1+o(1) r d r−1 . We also prove an analogous result for multidimensional rectangular grids. Our proofs exploit a connection between bootstrap percolation and a related process, known as weak saturation. In addition, we improve on the best known upper bound for the minimum size of a percolating set in the hypercube. In particular, when r = 3, we prove that the minimum cardinality of a percolating set in the d-dimensional hypercube is d(d+3) 6 + 1 for all d ≥ 3.

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Given graphs G and H and a positive integer q, say that G is q‐Ramsey for H, denoted G→(H)q, if every q‐coloring of the edges of G contains a monochromatic copy of H. The size‐Ramsey number truerˆ(H) of a graph H is defined to be truerˆ(H)=min{∣E(G)∣:G→(H)2}. Answering a question of Conlon, we prove that, for every fixed k, we have truerˆ(Pnk)=O(n), where Pnk is the kth power of the n‐vertex path Pn (ie, the graph with vertex set V(Pn) and all edges {u,v} such that the distance between u and v in Pn is at most k). Our proof is probabilistic, but can also be made constructive.

Given a set X, a collection F ⊆ P(X) is said to be k-Sperner if it does not contain a chain of length k + 1 under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. [11] conjectured that, if |X| is sufficiently large with respect to k, then the minimum size of a saturated k-Sperner system F ⊆ P(X) is 2 k−1 . We disprove this conjecture by showing that there exists ε > 0 such that for every k and |X| ≥ n 0 (k) there exists a saturated k-Sperner system F ⊆ P(X) with cardinality at most 2 (1−ε)k .A collection F ⊆ P(X) is said to be an oversaturated k-Sperner system if, for every S ∈ P(X) \ F, F ∪ {S} contains more chains of length k + 1 than F. Gerbner et al. [11] proved that, if |X| ≥ k, then the smallest such collection contains between 2 k/2−1 and O log k k 2 k elements. We show that if |X| ≥ k 2 + k, then the lower bound is best possible, up to a polynomial factor.

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