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22
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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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
SPECTRA OF UNIFORM HYPERGRAPHS
"... Abstract. We present a spectral theory of uniform hypergraphs that closely parallels Spectral Graph Theory. A number of recent developments building upon classical work has led to a rich understanding of “symmetric hyperdeterminants ” of hypermatrices, a.k.a. multidimensional arrays. Symmetric hyper ..."
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Abstract. We present a spectral theory of uniform hypergraphs that closely parallels Spectral Graph Theory. A number of recent developments building upon classical work has led to a rich understanding of “symmetric hyperdeterminants ” of hypermatrices, a.k.a. multidimensional arrays. Symmetric hyperdeterminants share many properties with determinants, but the context of multilinear algebra is substantially more complicated than the linear algebra required to address Spectral Graph Theory (i.e., ordinary matrices). Nonetheless, it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally. We apply this notion to the “adjacency hypermatrix” of a uniform hypergraph, and prove a number of natural analogues of basic results in Spectral Graph Theory. Open problems abound, and
Weak hypergraph regularity and linear hypergraphs
, 2009
"... We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any kuniform hypergraph H of positive uniform density contains all linear kuniform hypergraphs of a given size. More precisely, we show that for all integers ℓ ≥ k ≥ 2 and every d&g ..."
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We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any kuniform hypergraph H of positive uniform density contains all linear kuniform hypergraphs of a given size. More precisely, we show that for all integers ℓ ≥ k ≥ 2 and every d> 0 there exists ϱ> 0 for which the following holds: if H is a sufficiently large kuniform hypergraph with the property that the density of H induced on every vertex subset of size ϱn is at least d, then H contains every linear kuniform hypergraph F with ℓ vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph εregularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems.
The QuasiRandomness of Hypergraph Cut Properties
"... Let α1,..., αk satisfy ∑ i αi = 1 and suppose a kuniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A1,..., Ak of sizes α1n,..., αkn, the number of edges intersecting A1,..., Ak is (asymptotically) the number one would expect to find in a ..."
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Let α1,..., αk satisfy ∑ i αi = 1 and suppose a kuniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A1,..., Ak of sizes α1n,..., αkn, the number of edges intersecting A1,..., Ak is (asymptotically) the number one would expect to find in a random kuniform hypergraph. Can we then infer that H is quasirandom? We show that the answer is negative if and only if α1 = · · · = αk = 1/k. This resolves an open problem raised in 1991 by Chung and Graham [J. AMS ’91]. While hypergraphs satisfying the property corresponding to α1 = · · · = αk = 1/k are not necessarily quasirandom, we manage to find a characterization of the hypergraphs satisfying this property. Somewhat surprisingly, it turns out that (essentially) there is a unique non quasirandom hypergraph satisfying this property. The proofs combine probabilistic and algebraic arguments with results from the theory of association schemes. 1
Eigenvalues of nonregular linearquasirandom hypergraphs
, 2014
"... Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to coregular kuniform hypergraphs with loops. However, for k ≥ 3 no kuniform hypergraph is coregula ..."
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Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to coregular kuniform hypergraphs with loops. However, for k ≥ 3 no kuniform hypergraph is coregular. In this paper we remove the coregular requirement. Consequently, the characterization can be applied to kuniform hypergraphs; for example it is used in [19] to show that a construction of a kuniform hypergraph sequence is quasirandom.
Eigenvalues and Linear Quasirandom Hypergraphs
, 2013
"... 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 linear hypergraph quasirandomness of KohayakawaRödlSkokan and ConlonHànPersonSchacht and the spectral a ..."
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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 linear hypergraph quasirandomness of KohayakawaRödlSkokan and ConlonHànPersonSchacht and the spectral approach of FriedmanWigderson. For each of the quasirandom properties that are described, we define a largest and second largest eigenvalue. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of ConlonHànPersonSchacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of ChungGrahamWilson for graphs.
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.
QuasiRandom Hypergraphs Revisited
"... ABSTRACT: The quasirandom theory for graphs mainly focuses on a large equivalent class of graph properties each of which can be used as a certificate for randomness. For kgraphs (i.e., kuniform hypergraphs), an analogous quasirandom class contains various equivalent graph properties including th ..."
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ABSTRACT: The quasirandom theory for graphs mainly focuses on a large equivalent class of graph properties each of which can be used as a certificate for randomness. For kgraphs (i.e., kuniform hypergraphs), an analogous quasirandom class contains various equivalent graph properties including the kdiscrepancy property (bounding the number of edges in the generalized induced subgraph determined by any given (k − 1)graph on the same vertex set) as well as the kdeviation property (bounding the occurrences of “octahedron”, a generalization of 4cycle). In a 1990 paper (Chung, Random Struct Algorithms 1 (1990) 363382), a weaker notion of ldiscrepancy properties for kgraphs was introduced for forming a nested chain of quasirandom classes, but the proof for showing the equivalence of ldiscrepancy and ldeviation, for 2 ≤ l < k, contains an error. An additional parameter is needed in the definition of discrepancy, because of the rich and complex structure in hypergraphs. In this note, we introduce the notion of (l, s)discrepancy for kgraphs and prove that the equivalence of the (k, s)discrepancy and the sdeviation for 1 ≤ s ≤ k. We remark that this refined notion of discrepancy seems to point to a lattice structure in relating various quasirandom