Hereditary properties of combinatorial structures: posets and oriented graphs

A hereditary property of combinatorial structures is a collection of structures (e.g. graphs, posets) which is closed under isomorphism, closed under taking induced substructures (e.g. induced subgraphs), and contains arbitrarily large structures. Given a property P, we write P_n for the collection of distinct (i.e., non-isomorphic) structures in a property P with n vertices, and call the function n -> |P_n| the speed (or unlabelled speed) of P. Also, we write P^n for the collection of distinct labelled structures in P with vertices labelled 1,...,n, and call the function n -> |P^n| the labelled speed of P. The possible labelled speeds of a hereditary property of graphs have been extensively studied, and the aim of this paper is to investigate the possible speeds of other combinatorial structures, namely posets and oriented graphs. More precisely, we show that (for sufficiently large n), the labelled speed of a hereditary property of posets is either 1, or exactly a polynomial, or at least 2^n - 1. We also show that there is an initial jump in the possible unlabelled speeds of hereditary properties of posets, tournaments and directed graphs, from bounded to linear speed, and give a sharp lower bound on the possible linear speeds in each case.Comment: 26 pgs, no figure

Bootstrap percolation in high dimensions

In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A \subset V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vertices are infected. Our aim is to understand this process on the grid, [n]^d, for arbitrary functions n = n(t), d = d(t) and r = r(t), as t -> infinity. The main question is to determine the critical probability p_c([n]^d,r) at which percolation becomes likely, and to give bounds on the size of the critical window. In this paper we study this problem when r = 2, for all functions n and d satisfying d \gg log n. The bootstrap process has been extensively studied on [n]^d when d is a fixed constant and 2 \leq r \leq d, and in these cases p_c([n]^d,r) has recently been determined up to a factor of 1 + o(1) as n -> infinity. At the other end of the scale, Balogh and Bollobas determined p_c([2]^d,2) up to a constant factor, and Balogh, Bollobas and Morris determined p_c([n]^d,d) asymptotically if d > (log log n)^{2+\eps}, and gave much sharper bounds for the hypercube. Here we prove the following result: let \lambda be the smallest positive root of the equation \sum_{k=0}^\infty (-1)^k \lambda^k / (2^{k^2-k} k!) = 0, so \lambda \approx 1.166. Then (16\lambda / d^2) (1 + (log d / \sqrt{d})) 2^{-2\sqrt{d}} < p_c([2]^d,2) < (16\lambda / d^2) (1 + (5(log d)^2 / \sqrt{d})) 2^{-2\sqrt{d}} if d is sufficiently large, and moreover we determine a sharp threshold for the critical probability p_c([n]^d,2) for every function n = n(d) with d \gg log n.Comment: 51 pages, revised versio

Hereditary properties of partitions, ordered graphs and ordered hypergraphs

In this paper we use the Klazar-Marcus-Tardos method to prove that if a hereditary property of partitions P has super-exponential speed, then for every k-permutation pi, P contains the partition of [2k] with parts {i, pi(i) + k}, where 1 <= i <= k. We also prove a similar jump, from exponential to factorial, in the possible speeds of monotone properties of ordered graphs, and of hereditary properties of ordered graphs not containing large complete, or complete bipartite ordered graphs. Our results generalize the Stanley-Wilf Conjecture on the number of n-permutations avoiding a fixed permutation, which was recently proved by the combined results of Klazar and of Marcus and Tardos. Our main results follow from a generalization to ordered hypergraphs of the theorem of Marcus and Tardos.Comment: 25 pgs, no figure

Hereditary properties of tournaments

A collection of unlabelled tournaments P is called a hereditary property if it is closed under isomorphism and under taking induced sub-tournaments. The speed of P is the function n -> |P_n|, where P_n = {T \in P : |V(T)| = n}. In this paper, we prove that there is a jump in the possible speeds of a hereditary property of tournaments, from polynomial to exponential speed. Moreover, we determine the minimal exponential speed, |P_n| = c^(n + o(n)), where c = 1.47... is the largest real root of the polynomial x^3 = x^2 + 1, and the unique hereditary property with this speed.Comment: 28 pgs, 2 figures, submitted November 200

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

The sharp threshold for bootstrap percolation in all dimensions

In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the d-dimensional grid $[n]^d$. The elements of the set A are usually chosen independently, with some density p, and the main question is to determine $p_c([n]^d,r)$, the density at which percolation (infection of the entire vertex set) becomes likely. In this paper we prove, for every pair $d \ge r \ge 2$, that there is a constant L(d,r) such that $p_c([n]^d,r) = [(L(d,r) + o(1)) / log_(r-1) (n)]^{d-r+1}$ as $n \to \infty$, where $log_r$ denotes an r-times iterated logarithm. We thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. Moreover, we determine L(d,r) for every pair (d,r).Comment: 37 pages, sketch of the proof added, to appear in Trans. of the AM