In this paper we study in complete generality the family of two-state, deterministic, monotone, local, homogeneous cellular automata in Z with random initial configurations. Formally, we are given aâ€¦ (More)

In recent breakthrough results, Saxtonâ€“Thomason and Baloghâ€“Morrisâ€“Samotij have developed powerful theories of hypergraph containers. These theories have led to a large number of new results onâ€¦ (More)

Given a graph property P, it is interesting to determine the typical structure of graphs that satisfy P. In this paper, we consider monotone properties, that is, properties that are closed underâ€¦ (More)

We study limits of convergent sequences of string graphs, that is, graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior ofâ€¦ (More)

In r-neighbor bootstrap percolation on the vertex set of a graph G, vertices are initially infected independently with some probability p. At each time step, the infected set expands by infecting allâ€¦ (More)

We study maximal Kr+1-free graphs G of almost extremal sizeâ€”typically, e(G) = ex(n,Kr+1) âˆ’ O(n). We show that any such graph G must have a large amount of â€˜symmetryâ€™: in particular, all but very fewâ€¦ (More)