We show that if a sequence of dense graphs G n has the property that for every fixed graph F , the density of copies of F in G n tends to a limit, then there is a natural "limit object," namely a symmetric measurable function W : [0, 1] 2 → [0, 1]. This limit object determines all the limits of subgraph densities. Conversely, every such function arises as a limit object. We also characterize graph parameters that are obtained as limits of subgraph densities by the "reflection positivity" property.Along the way we introduce a rather general model of random graphs, which seems to be interesting on its own right.

Szemerédi's regularity lemma is a fundamental tool in graph theory: it has many applications to extremal graph theory, graph property testing, combinatorial number theory, etc. The goal of this paper is to point out that Szemerédi's lemma can be thought of as a result in analysis. We show three different analytic interpretations.

We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2.41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2.

In this paper we develop a measure-theoretic method to treat problems in hypergraph theory.Our central theorem is a correspondence principle between three objects: An increasing hypergraph sequence, a measurable set in an ultraproduct space and a measurable set in a finite dimensional Lebesgue space. Using this correspondence principle we build up the theory of dense hypergraphs from scratch. Along these lines we give new proofs for the Hypergraph Removal Lemma, the Hypergraph Regularity Lemma, the Counting Lemma and the Testability of Hereditary Hypergraph Properties. We prove various new results including a strengthening of the Regularity Lemma and an Inverse Counting Lemma. We also prove the equivalence of various notions for convergence of hypergraphs and we construct limit objects for such sequences. We prove that the limit objects are unique up to a certain family of measure preserving transformations. As our main tool we study the integral and measure theory on the ultraproduct of finite measure spaces which is interesting on its own right. *

The colored neighborhood metric for sparse graphs was introduced by Bollobás and Riordan [8]. The corresponding convergence notion refines a convergence notion introduced by Benjamini and Schramm [6]. We prove that even in this refined sense, the limit of a convergent graph sequence (with uniformly bounded degree) can be represented by a graphing. We study various topics related to this convergence notion such as: Bernoulli graphings, factor of i.i.d. processes and hyperfiniteness.

We investigate families of graphs and graphons (graph limits) that are determined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by finitely many subgraph densities. Generalizing results of Turán, Erdős-Simonovits and Chung-Graham-Wilson, we construct numerous finitely forcible graphons. Most of these fall into two categories: one type has an algebraic structure and the other type has an iterated (fractal-like) structure. We also give some necessary conditions for forcibility, which imply that finitely forcible graphons are "rare", and exhibit simple and explicit non-forcible graphons.

We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon

We define a distance of two graphs that reflects the closeness of both local and global properties. We also define convergence of a sequence of graphs, and show that a graph sequence is convergent if and only if it is Cauchy in this distance. Every convergent graph sequence has a limit in the form of a symmetric measurable function in two variables. We use these notions of distance and graph limits to give a general theory for parameter testing. As examples, we provide short proofs of the testability of MaxCut and the recent result of Alon and Shapira about the testability of hereditary graph properties.