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38
The counting lemma for regular kuniform hypergraphs
, 2004
"... Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εregular of ..."
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Cited by 108 (14 self)
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Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εregular of density d for ℓ 1 ≤ i < j ≤ ℓ, then G contains (1 ± fℓ(ε))d
A variant of the hypergraph removal lemma
, 2006
"... Abstract. Recent work of Gowers [10] and Nagle, Rödl, Schacht, and Skokan [15], [19], [20] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [26] and FurstenbergKatznelson [7] concerning onedimensional and multidimensional arithmetic progressions respecti ..."
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Cited by 77 (7 self)
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Abstract. Recent work of Gowers [10] and Nagle, Rödl, Schacht, and Skokan [15], [19], [20] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [26] and FurstenbergKatznelson [7] concerning onedimensional and multidimensional arithmetic progressions respectively. In this paper we shall give a selfcontained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper [29] to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes. 1.
Regular partitions of hypergraphs: Regularity Lemmas
 COMBIN. PROBAB. COMPUT
, 2007
"... Szemerédi’s regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and authors and obtain a stronger and more “user friendly” regularity lemma for hypergraphs. ..."
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Cited by 29 (1 self)
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Szemerédi’s regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and authors and obtain a stronger and more “user friendly” regularity lemma for hypergraphs.
The Gaussian primes contain arbitrarily shaped constellations
 J. Analyse Math
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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.
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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Cited by 21 (2 self)
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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
D.: Embeddings and Ramsey numbers of sparse kuniform hypergraphs, Combinatorica 29
, 2009
"... Abstract. Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [5, 19] the same result was proved for 3uniform hypergraphs. Here we extend this result to kuniform hypergraphs for any integer k ≥ 3. As in the 3uni ..."
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Cited by 19 (4 self)
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Abstract. Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [5, 19] the same result was proved for 3uniform hypergraphs. Here we extend this result to kuniform hypergraphs for any integer k ≥ 3. As in the 3uniform case, the main new tool which we prove and use is an embedding lemma for kuniform hypergraphs of bounded maximum degree into suitable kuniform ‘quasirandom ’ hypergraphs.
Regular partitions of hypergraphs: Counting Lemmas
 COMBIN. PROBAB. COMPUT
"... We continue the study of regular partitions of hypergraphs. In particular we obtain corresponding counting lemmas for the regularity lemmas for hypergraphs from [Regular partitions of hypergraphs: Regularity Lemmas, Combin. Probab. Comput., to appear]. ..."
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Cited by 16 (3 self)
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We continue the study of regular partitions of hypergraphs. In particular we obtain corresponding counting lemmas for the regularity lemmas for hypergraphs from [Regular partitions of hypergraphs: Regularity Lemmas, Combin. Probab. Comput., to appear].
Hamilton ℓcycles in uniform hypergraphs
 JOURNAL OF COMBINATORIAL THEORY. SERIES A
"... We say that a kuniform hypergraph C is an ℓcycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely ℓ vertices. We prove th ..."
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Cited by 12 (3 self)
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We say that a kuniform hypergraph C is an ℓcycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely ℓ vertices. We prove that if 1 ≤ ℓ < k and k − ℓ does not divide k then any kuniform hypergraph on n vertices with minimum degree at least nd k k− ` e(k−`) +o(n) contains a Hamilton ℓcycle. This confirms a conjecture of Hàn and Schacht. Together with results of Rödl, Ruciński and Szemerédi, our result asymptotically determines the minimum degree which forces an `cycle for any ` with 1 ≤ ℓ < k.
Bipartite Subgraphs and Quasirandomness
"... Abstract. We say that a family of graphs G = {Gn: n ≥ 1} is pquasirandom, 0 < p < 1, if it shares typical properties of the random graph G(n, p); for a definition, see below. We denote by Q w � (p) the class of all graphs H for which e(Gn) ≥ (1 + and the number of not necessarily induced la ..."
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Abstract. We say that a family of graphs G = {Gn: n ≥ 1} is pquasirandom, 0 < p < 1, if it shares typical properties of the random graph G(n, p); for a definition, see below. We denote by Q w � (p) the class of all graphs H for which e(Gn) ≥ (1 + and the number of not necessarily induced labeled copies of H in Gn is at o(1))p �n 2 most (1 + o(1))p e(H) n v(H) imply that G is pquasirandom. In this note, we show that all complete bipartite graphs Ka,b, a, b ≥ 2, belong to Q w (p) for all 0 < p < 1. 1. Notation We start with fixing notation. For positive integers k, n and a real number x, we set [n] = {1,..., n} and (x)k = x(x − 1) × · · · × (x − k + 1). Given a graph G with vertex set V (G) and edge set E(G), v(G) stands for V (G)  and e(G) for E(G). Furthermore, for a subset X of V (G), G[X] denotes the subgraph induced by the vertices of X, and e(X) denotes the number of edges of G[X]. Given a vertex x ∈ V (G), NG(x) is the set of all vertices adjacent to x and, similarly, for a subset X of V (G), NG(X) denotes the set of all vertices adjacent to every vertex in X. Clearly, NG(X) = � x∈X NG(x). We also put