Balázs Szegedy

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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] → [0, 1]. This limit object determines all the limits of subgraph densities. Conversely, every such function arises as a(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)
The motivation of this paper comes from statistical physics as well as from combinatorics and topology. The general setup in statistical mechanics can be outlined as follows. Let G be a graph and let C be a finite set of “states” or “colors”. We think of G as a crystal in which either the edges or the vertices are regarded as “sites” which can have states(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)
Let A = {a1, . . . , ak} and B = {b1, . . . , bk} be two subsets of an Abelian group G, k ≤ |G|. Snevily conjectured that, when G is of odd order, there is a permutation π ∈ Sk such that the sums ai+bπ(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| elements,(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, Erdős–Simonovits and Chung–Graham–Wilson, we construct(More)
We study generalizations of the “contraction-deletion” relation of the Tutte polynomial, and other similar simple operations, to other graph parameters. The question can be set in the framework of graph algebras introduced by Freedman, Lovász and Schrijver in [2], and it relates to their behavior under basic graph operations like contraction and(More)
In this note we observe that in the hyper-graph removal lemma the edge removal can be done in a way that the symmetries of the original hyper-graph remain preserved. As an application we prove the following generalization of Szemerédi’s Theorem on arithmetic progressions. If in an Abelian group A there are sets S1, S2 . . . , St such that the number of(More)