353 research outputs found
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 . The elements of
the set A are usually chosen independently, with some density p, and the main
question is to determine , the density at which percolation
(infection of the entire vertex set) becomes likely.
In this paper we prove, for every pair , that there is a
constant L(d,r) such that as , where 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
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