On the Typical Structure of Graphs in a Monotone Property

Given a graph property $\mathcal{P}$, it is interesting to determine the typical structure of graphs that satisfy $\mathcal{P}$. In this paper, we consider monotone properties, that is, properties that are closed under taking subgraphs. Using results from the theory of graph limits, we show that if $\mathcal{P}$ is a monotone property and $r$ is the largest integer for which every $r$-colorable graph satisfies $\mathcal{P}$, then almost every graph with $\mathcal{P}$ is close to being a balanced $r$-partite graph.Comment: 5 page

An extension of Tur\'an's Theorem, uniqueness and stability

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique extremal graph. For $(r-1)|M|<n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.Comment: 12 pages, 1 figure; outline of the proof added and other referee's comments incorporate

Graph properties, graph limits and entropy

We study the relation between the growth rate of a graph property and the entropy of the graph limits that arise from graphs with that property. In particular, for hereditary classes we obtain a new description of the colouring number, which by well-known results describes the rate of growth. We study also random graphs and their entropies. We show, for example, that if a hereditary property has a unique limiting graphon with maximal entropy, then a random graph with this property, selected uniformly at random from all such graphs with a given order, converges to this maximizing graphon as the order tends to infinity.Comment: 24 page

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 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