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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)
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)