We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. To develop a mathematical theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs, which has emerged over the last decade. The theory has rich connections with other approaches to the study of large networks, such as "property testing" in computer science and regularity partition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connections with other parts of mathematics (classical and non-classical, like probability theory, measure theory, tensor algebras, and semidefinite optimization). This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the theory of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits. This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future. --Persi Diaconis, Stanford University This book is a comprehensive study of the active topic of graph limits and an updated account of its present status. It is a beautiful volume written by an outstanding mathematician who is also a great expositor. --Noga Alon, Tel Aviv University, Israel Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovasz's book exemplifies this phenomenon. This book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory. --Terence Tao, University of California, Los Angeles, CA Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovasz's position as the main architect of this rapidly developing theory. The book is a must for combinatorialists, network theorists, and theoretical computer scientists alike. --Bela Bollobas, Cambridge University, UK

Large Networks and Graph Limits

In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Property Testing and its connection to Learning and Approximation

Modern Graph Theory

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values.

Small Ramsey Numbers

Szemeredi's Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an e-partite graph with V (G) = V1 ∪ … ∪ Ve and sVis = n for all i ∈ [e], and all pairs (Vi, Vj) are e-regular of density d for 1 ≤ i ≤ j ≤ e and e L d, then G contains $(1\pm f_{\ell}(\varepsilon))d^{\ell \choose 2}\times n^{\ell}$ cliques Ke, where fe(e) → 0 as e → 0.Recently, Rodl and Skokan generalized Szemeredi's Regularity Lemma from graphs to k-uniform hypergraphs for arbitrary k ≥ 2. In this paper we prove a Counting Lemma accompanying the Rodl–Skokan hypergraph Regularity Lemma. Similar results were independently obtained by Gowers.Such results give combinatorial proofs to the density result of Szemeredi and some of the density theorems of Furstenberg and Katznelson. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

The counting lemma for regular k-uniform hypergraphs

We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemer edi’s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions, and we determine the threshold for Tur an-type problems for random graphs and hypergraphs. In particular, we verify a conjecture of

Extremal results for random discrete structures

We study sufficient $\ell$-degree ($1\leq\ell<k$) conditions for the appearance of perfect and nearly perfect matchings in $k$-uniform hypergraphs. In particular, we obtain a minimum vertex degree condition ($\ell=1$) for 3-uniform hypergraphs, which is approximately tight, by showing that every 3-uniform hypergraph on $n$ vertices with minimum vertex degree at least $(5/9+o(1))\binom{n}{2}$ contains a perfect matching.

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On Perfect Matchings in Uniform Hypergraphs with Large Minimum Vertex Degree

Szemeredi's regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and the authors and obtain a stronger and more ‘user-friendly’ regularity lemma for hypergraphs.

/pdf/regular-partitions-of-hypergraphs-regularity-lemmas-4bwm2dhrfk.pdf

Regular Partitions of Hypergraphs: Regularity Lemmas

In this paper we prove the following conjecture by Bollobas and Komlos: For every γ > 0 and integers r ≥ 1 and Δ, there exists β > 0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least ((r − 1)/r + γ)n and H is an r-chromatic graph with n vertices, bandwidth at most β n and maximum degree at most Δ, then G contains a copy of H.

/pdf/proof-of-the-bandwidth-conjecture-of-bollobas-and-komlos-2lwevhwwa8.pdf

Mathias Schacht

Papers

The counting lemma for regular k-uniform hypergraphs

Extremal results for random discrete structures

On Perfect Matchings in Uniform Hypergraphs with Large Minimum Vertex Degree

Regular Partitions of Hypergraphs: Regularity Lemmas

Proof of the bandwidth conjecture of Bollobás and Komlós