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We show that if a sequence of dense graphs Gn has the property that for every fixed graph F , the density of copies of F in Gn 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(More)
The <i>exponent of matrix multiplication</i> is the smallest real number &#969; such that for all &#949;&gt;0, <i>O</i>(n<sup>&#969;+&#949;</sup>) arithmetic operations suffice to multiply two <i>n&#215;n</i> matrices. The standard algorithm for matrix multiplication shows that &#969;&#8804;3. Strassen's remarkable result [5] shows that &#969;&#8804;2.81,(More)
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.
Let A = {a 1 ,. .. , a k } and B = {b 1 ,. .. , b k } be two subsets of an Abelian group G, k ≤ |G|. Snevily conjectured that, when G is of odd order, there is a permutation π ∈ S k such that the sums a i + b π(i) , 1 ≤ i ≤ k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even when A is a sequence of k < |G|(More)
We investigate families of graphs and graphons (graph limits) that are defined 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, Erd˝ os–Simonovits and Chung–Graham–Wilson, we(More)
In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplica-tive and reflection positive graph parameters. In this paper we show that each of these(More)
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(More)