In this paper we study in complete generality the family of two-state, deterministic, monotone, local, homogeneous cellular automata in Z d with random initial configurations. Formally, we are given a set U = {X1, . . . , Xm} of finite subsets of Z d \ {0}, and an initial set A0 ⊂ Z d of 'infected' sites, which we take to be random according to the product measure with density p. At time t ∈ N, the set of infected sites At is the union of At−1 and the set of all x ∈ Z d such that x + X ∈ At−1 for some X ∈ U. Our model may alternatively be thought of as bootstrap percolation on Z d with arbitrary update rules, and for this reason we call it U-bootstrap percolation.In two dimensions, we give a classification of U-bootstrap percolation models into three classes -supercritical, critical and subcritical -and we prove results about the phase transitions of all models belonging to the first two of these classes. More precisely, we show that the critical probability for percolation on (Z/nZ) 2 is (log n) −Θ(1) for all models in the critical class, and that it is n −Θ(1) for all models in the supercritical class.The results in this paper are the first of any kind on bootstrap percolation considered in this level of generality, and in particular they are the first that make no assumptions of symmetry. It is the hope of the authors that this work will initiate a new, unified theory of bootstrap percolation on Z d .A considerable amount is now known about p c (G, r) in the case of lattice graphs: a graph G is a (d-dimensional) lattice graph if it is (isomorphic to) a (not necessarily planar) translation invariant locally finite graph with vertex set Z d . (Equivalently, G is a lattice graph if (there is an isomorphism between V (G) and Z d under which) there exists a finite symmetric set X ⊂ Z d \ {0} such that the neighbourhood of any vertex x is the set x + X.) The most natural lattice graph, and the one that has attracted the greatest amount of study, is the nearest neighbour graph on Z d . Indeed, the first rigorous result in the field of bootstrap percolation, which was proved by van Enter [20], was that p c (Z 2 , 2) = 0. Schonmann [18] later showed that p c (Z d , r) is equal to 0 if r d and 1 otherwise. Aizenman and Lebowitz [1] were the first to recognize that there is much more to say about the model in the context of finite subgraphs of lattice graphs, and they proved that p c ([n] d , 2) = Θ(1)/(log n) d−1 . Holroyd [15] went considerably further in the case d = r = 2, proving that p c ([n] 2 , 2) = (1 + o(1))π 2 /18 log n. Vast generalizations of these results on [n] d for general d and r were obtained in the weak sense by Cerf and Cirillo [9] and Cerf and Manzo [10], and in the sharp sense by Balogh, Bollobás and Morris [5] and Balogh, Bollobás, Duminil-Copin and Morris [4].Bootstrap percolation has been studied on a number of other lattice graphs, not only on Z d and [n] d ; for a small selection of these results, see [3, 6,7,8,13, 14,17]. In particular, Gravner and Griffeath [13, 14] studied r-neighbour boo...

In breakthrough results, Saxton‐Thomason and Balogh‐Morris‐Samotij developed powerful theories of hypergraph containers. In this paper, we explore some consequences of these theories. We use a simple container theorem of Saxton‐Thomason and an entropy‐based framework to deduce container and counting theorems for hereditary properties of k‐colorings of very general objects, which include both vertex‐ and edge‐colorings of general hypergraph sequences as special cases. In the case of sequences of complete graphs, we further derive characterization and transference results for hereditary properties in terms of their stability families and extremal entropy. This covers within a unified framework a great variety of combinatorial structures, some of which had not previously been studied via containers: directed graphs, oriented graphs, tournaments, multigraphs with bounded multiplicity, and multicolored graphs among others. Similar results were recently and independently obtained by Terry.

In r-neighbour 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 all uninfected vertices that have at least r infected neighbours. When p is close to 1, we study the distribution of the time at which all vertices become infected. Given t = t(n) = o(log n/ log log n), we prove a sharp threshold result for the probability that percolation occurs by time t in d-neighbour bootstrap percolation on the d-dimensional discrete torus T d n. Moreover, we show that for certain ranges of p = p(n), the time at which percolation occurs is concentrated either on a single value or on two consecutive values. We also prove corresponding results for the modified d-neighbour rule.

The dimension of a partially-ordered set (poset), introduced by Dushnik and Miller (1941), has been studied extensively in the literature. Recently, Ueckerdt (2016) proposed a variation called local dimension which makes use of partial linear extensions. While local dimension is bounded above by dimension, they can be arbitrarily far apart as the dimension of the standard example is n while its local dimension is only 3.Hiraguchi (1955) proved that the maximum dimension of a poset of order n is n/2. However, we find a very different result for local dimension, proving a bound of Θ(n/ log n). This follows from connections with covering graphs using difference graphs which are bipartite graphs whose vertices in a single class have nested neighborhoods.We also prove that the local dimension of the n-dimensional Boolean lattice is Ω(n/ log n) and make progress toward resolving a version of the removable pair conjecture for local dimension.2010 Mathematics Subject Classification. 06A07,05C70.

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 of a sequence of random string graphs. We also prove similar results for several related graph classes.