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11
An approximate version of Sidorenko’s conjecture
 Geom. Funct. Anal
"... A beautiful conjecture of ErdősSimonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent ana ..."
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Cited by 23 (7 self)
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A beautiful conjecture of ErdősSimonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs. 1
Weak quasirandomness for uniform hypergraphs
, 2009
"... We study quasirandom properties of kuniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known ChungGrahamWilson theorem fo ..."
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Cited by 22 (8 self)
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We study quasirandom properties of kuniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known ChungGrahamWilson theorem for quasirandom graphs. Moreover, let Kk be the complete graph on k vertices and M(k) the line graph of the graph of the kdimensional hypercube. We will show that the pair of graphs (Kk, M(k)) has the property that if the number of copies of both Kk and M(k) in another graph G are as expected in the random graph of density d, then G is quasirandom (in the sense of the ChungGrahamWilson theorem) with density close to d.
An algorithmic version of the hypergraph regularity method (extended abstract
 Proceedings of the IEEE Symposium on Foundations of Computer Science
, 2005
"... Abstract. Extending the Szemerédi Regularity Lemma for graphs, P. Frankl and V. Rödl [14] established a 3graph Regularity Lemma guaranteeing that all large triple systems Gn admit bounded partitions of their edge sets, most classes of which consist of regularly distributed triples. Many applicati ..."
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Cited by 9 (6 self)
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Abstract. Extending the Szemerédi Regularity Lemma for graphs, P. Frankl and V. Rödl [14] established a 3graph Regularity Lemma guaranteeing that all large triple systems Gn admit bounded partitions of their edge sets, most classes of which consist of regularly distributed triples. Many applications of this lemma require a companion Counting Lemma [30], allowing one to find and enumerate subhypergraphs of a given isomorphism type in a “dense and regular ” environment created by the 3graph Regularity Lemma. Combined applications of these lemmas are known as the 3graph Regularity Method. In this paper, we provide an algorithmic version of the 3graph Regularity Lemma which, as we show, is compatible with a Counting Lemma. We also discuss some applications. 1.
The Poset of Hypergraph Quasirandomness
, 2012
"... Chung and Graham began the systematic study of hypergraph quasirandom properties soon after the foundational results of Thomason and ChungGrahamWilson on quasirandom graphs. One feature that became apparent in the early work on hypergraph quasirandomness is that properties that are equivalent for ..."
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Cited by 4 (2 self)
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Chung and Graham began the systematic study of hypergraph quasirandom properties soon after the foundational results of Thomason and ChungGrahamWilson on quasirandom graphs. One feature that became apparent in the early work on hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and a fundamental open problem that has emerged is to determine the relationship between these quasirandom properties. We completely determine the poset of implications between essentially all hypergraph quasirandom properties that have been studied in the literature. This answers a recent question of Chung, and in some sense completes the project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990’s. 1
Eigenvalues and Quasirandom Hypergraphs
, 2012
"... Let p(k) denote the partition function of k. For each k ≥ 2, we describe a list of p(k) − 1 quasirandom properties that a kuniform hypergraph can have. Our work connects previous notions on hypergraph quasirandomness, beginning with the early work of Chung and Graham and FranklRödl related to str ..."
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Cited by 4 (0 self)
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Let p(k) denote the partition function of k. For each k ≥ 2, we describe a list of p(k) − 1 quasirandom properties that a kuniform hypergraph can have. Our work connects previous notions on hypergraph quasirandomness, beginning with the early work of Chung and Graham and FranklRödl related to strong hypergraph regularity, the spectral approach of FriedmanWigderson, and more recent results of KohayakawaRödlSkokan and ConlonHànPersonSchacht on weak hypergraph regularity and its relation to counting linear hypergraphs. For each of the quasirandom properties that are described, we define a hypergraph eigenvalue analogous to the graph case and a hypergraph extension of a graph cycle of even length whose count determines if a hypergraph satisfies the property. This answers a question of Conlon et al. Our work can be viewed as an extension to hypergraphs of the seminal results of ChungGrahamWilson for graphs. Our results yield the following applications. First, motivated by Sidorenko’s Conjecture on the minimum homomorphism density of bipartite graphs in arbitrary graphs, we show that an analog of the conjecture for hypergraphs holds for a variety of hypergraph cycles. These are the first infinite families of hypergraphs with minimum degree two where this has been verified. Second, we give an efficient certification algorithm for hypergraph quasirandomnes which leads to an efficient strong refutation algorithm for random kSAT. For nvertex, kuniform hypergraphs with k ≥ 4 and at least n k/2+ √ k edges, we provide an algorithm with running time O(n kω polylog n) that certifies quasirandomness for almost all hypergraphs. This improves the previous best running time for such certification due to CojaOghlanCooperFrieze and HánPersonSchacht, in addition to also certifying a stronger quasirandom property than these previous results.
Note on forcing pairs
, 2011
"... The notion of forcing pairs is located in the study of quasirandom graphs. Roughly speaking, a pair of graphs (F, F ′ ) is called forcing if the following holds: Suppose for a sequence of graphs (Gn) there is a p> 0 such that the number of copies of F and the number of copies of F ′ in every gra ..."
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Cited by 1 (1 self)
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The notion of forcing pairs is located in the study of quasirandom graphs. Roughly speaking, a pair of graphs (F, F ′ ) is called forcing if the following holds: Suppose for a sequence of graphs (Gn) there is a p> 0 such that the number of copies of F and the number of copies of F ′ in every graph Gn of the sequence (Gn) is approximately the same as the expected value in the random graph G(n, p), then the sequence of graphs (Gn) is quasirandom in the sense of Chung, Graham and Wilson. We describe a construction which, given any graph F with at least one edge, yields a graph F ′ such that (F, F ′ ) forms a forcing pair.
On the Density of a Graph and its Blowup
"... The theorem of Chung, Graham, and Wilson on quasirandom graphs asserts that of all graphs with edge density p, the random graph G(n, p) contains the smallest density of copies of Kt,t, the complete bipartite graph of size 2t. Since Kt,t is a tblowup of an edge, the following intriguing open questi ..."
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The theorem of Chung, Graham, and Wilson on quasirandom graphs asserts that of all graphs with edge density p, the random graph G(n, p) contains the smallest density of copies of Kt,t, the complete bipartite graph of size 2t. Since Kt,t is a tblowup of an edge, the following intriguing open question arises: Is it true that of all graphs with triangle density p3, the random graph G(n, p) contains the smallest density of Kt,t,t, which is the tblowup of a triangle? Our main result gives an indication that the answer to the above question is positive by showing that for some blowup, the answer must be positive. More formally we prove that if G has triangle density p3, then there is some 2 ≤ t ≤ T (p) for which the density of Kt,t,t in G is at least p (3+o(1))t2, which (up to the o(1) term) equals the density of Kt,t,t in G(n, p). We also consider the analogous question on skewed blowups, showing that somewhat surprisingly, the behavior there is different. We also raise several conjectures related to these problems and discuss some applications to other areas. 1
TREEMINIMAL GRAPHS ARE ALMOST REGULAR
"... Abstract. For all fixed trees T and any graph G we derive a counting formula for the number N̂T (G) of homomorphisms from T to G in terms of the degree sequence of G. As a consequence we obtain that any nvertex graph G with edge density p = p(n) n−1/(t−2), which contains at most (1 + o(1))pt−1nt c ..."
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Abstract. For all fixed trees T and any graph G we derive a counting formula for the number N̂T (G) of homomorphisms from T to G in terms of the degree sequence of G. As a consequence we obtain that any nvertex graph G with edge density p = p(n) n−1/(t−2), which contains at most (1 + o(1))pt−1nt copies of some fixed tree T with t ≥ 3 vertices must be almost regular, i.e., v∈V (G) deg(v) − pn  = o(pn2).
Sparse Pseudorandom Objects
, 2010
"... It has been known for a long time that many mathematical objects can be naturally decomposed into a ‘pseudorandom’, chaotic part and/or a highly organized ‘periodic ’ component. Theorems or heuristics of this type have been used in combinatorics, harmonic analysis, dynamical systems and other parts ..."
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It has been known for a long time that many mathematical objects can be naturally decomposed into a ‘pseudorandom’, chaotic part and/or a highly organized ‘periodic ’ component. Theorems or heuristics of this type have been used in combinatorics, harmonic analysis, dynamical systems and other parts of mathematics for many years, but a number of results related to such ‘structural ’ theorems emerged only in the last decades. A seminal example of such a structural theorem in discrete mathematics is Szeméredi’s Regularity Lemma, which was discovered by Szeméredi in the midseventies when he proved his famous result on arithmetic progressions in dense subsets of natural numbers. It states that the set of edges of any dense graph can be ‘nearly decomposed ’ into ‘pseudorandom ’ bipartite graphs. The Regularity Lemma has long been recognised as one of the most powerful tools of modern graph theory. The aim of the meeting was to follow this structural theme and investigate structural results for sparse combinatorial objects. The meeting brought together a number of experts in the area together with several junior researchers and PhD students. 2 Presentations and Discussions Each presenter described recent developments on a particular topic, outlined some of the main related open problems, and led an interactive discussion on these results and problems. The topics addressed were as follows. 2.1 Extremal problems for random discrete structures (M. Schacht) We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemerédi’s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we determine the threshold for Turántype problems for random graphs and hypergraphs. In particular, we verify a conjecture of Kohayakawa, Łuczak, and Rödl for Turántype problems in random graphs. Similar results were obtained by Conlon and Gowers. 2.2 Extremal Graph Theory – the Regularity Lemma Revisited (T. Łuczak) For a graph H and natural numbers k and n let us define the parameter ν (k) χ (H, n) [ν (k) τ (H, n)] as the smallest a such that each Hfree graph G with n vertices and the minimum degree δ(G) ≥ an can be homomorphically mapped to Kk [some Hfree graph F on k vertices]. The behavior of these two parameters
Treeminimal graphs are almost regular
"... For all fixed trees T and any graph G we derive a counting formula for the number ˆ NT (G) of homomorphisms from T to G in terms of the degree sequence of G. As a consequence we obtain that any nvertex graph G with edge density p = p(n) ≫ n −1/(t−2) , which contains at most (1 + o(1))p t−1 n t co ..."
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For all fixed trees T and any graph G we derive a counting formula for the number ˆ NT (G) of homomorphisms from T to G in terms of the degree sequence of G. As a consequence we obtain that any nvertex graph G with edge density p = p(n) ≫ n −1/(t−2) , which contains at most (1 + o(1))p t−1 n t copies of some fixed tree T with t ≥ 3 vertices must be almost regular, i.e., ∑ v∈V (G)  deg(v) − pn  = o(pn2).