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- László Lovász, Balázs Szegedy
- J. Comb. Theory, Ser. B
- 2006

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)

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.

- Henry Cohn, Robert D. Kleinberg, Balázs Szegedy, Christopher Umans
- 46th Annual IEEE Symposium on Foundations of…
- 2005

The <i>exponent of matrix multiplication</i> is the smallest real number ω such that for all ε>0, <i>O</i>(n<sup>ω+ε</sup>) arithmetic operations suffice to multiply two <i>n×n</i> matrices. The standard algorithm for matrix multiplication shows that ω≤3. Strassen's remarkable result [5] shows that ω≤2.81,… (More)

- BALÁZS SZEGEDY
- 2007

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)

- László Lovász, Balázs Szegedy
- J. Comb. Theory, Ser. B
- 2011

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 show that an important recent result of Alon and Shapira on testing hereditary graph properties can be derived from the existence of a limit object for convergent graph sequences.

- László Lovász, Balázs Szegedy
- Journal of Graph Theory
- 2009

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)

- Balázs Szegedy
- 2008

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)