Long running times for hypergraph bootstrap percolation

Consider the hypergraph bootstrap percolation process in which, given a fixed r-uniform hypergraph H and starting with a given hypergraph G0, at each step we add to G0 all edges that create a new copy of H. We are interested in maximising the number of steps that this process takes before it stabilises. For the case where H=Kr+1(r) with r≥3, we provide a new construction for G0 that shows that the number of steps of this process can be of order Θ(nr). This answers a recent question of Noel and Ranganathan. To demonstrate that different running times can occur, we also prove that, if H is K4(3) minus an edge, then the maximum possible running time is 2n−⌊log2(n−2)⌋−6. However, if H is K5(3) minus an edge, then the process can run for Θ(n3) steps

Majority bootstrap percolation on the hypercube

In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain infected forever. Percolation occurs if eventually every vertex is infected. The elements of the set of initially infected vertices, A \subset V(G), are normally chosen independently at random, each with probability p, say. This process has been extensively studied on the sequence of torus graphs [n]^d, for n = 1,2,..., where d = d(n) is either fixed or a very slowly growing function of n. For example, Cerf and Manzo showed that the critical probability is o(1) if d(n) < log*(n), i.e., if p = p(n) is bounded away from zero then the probability of percolation on [n]^d tends to one as n goes to infinity. In this paper we study the case when the growth of d to infinity is not excessively slow; in particular, we show that the critical probability is 1/2 + o(1) if d > (loglog(n))^2 logloglog(n), and give much stronger bounds in the case that G is the hypercube, [2]^d.Comment: 44 pgs, no figures, submitted Feb 200

Combinatorics and Probability

For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices