The number of sub-matrics of a given type in a Hadamard matrix and related results
Journal of Combinatorial Theory Series A
Discrete Mathematics
The algorithmic aspects of the regularity lemma
Journal of Algorithms
A Fast Approximation Algorithm for Computing theFrequencies of Subgraphs in a Given Graph
SIAM Journal on Computing
Ramsey properties of random hypergraphs
Journal of Combinatorial Theory Series A
Constructive Quasi-Ramsey Numbers and Tournament Ranking
SIAM Journal on Discrete Mathematics
Extremal problems on set systems
Random Structures & Algorithms
Integer and fractional packings in dense 3-uniform hypergraphs
Random Structures & Algorithms
The regularity lemma and approximation schemes for dense problems
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Regularity properties for triple systems
Random Structures & Algorithms
Hereditary Properties of Triple Systems
Combinatorics, Probability and Computing
An Algorithmic Version of the Hypergraph Regularity Method
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
The counting lemma for regular k-uniform hypergraphs
Random Structures & Algorithms
Loose Hamilton cycles in 3-uniform hypergraphs of high minimum degree
Journal of Combinatorial Theory Series B
Hypergraph regularity and quasi-randomness
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
On Computing the Frequencies of Induced Subhypergraphs
SIAM Journal on Discrete Mathematics
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Szemerédi's Regularity Lemma is a well-known and powerful tool in modern graph theory. This result led to a number of interesting applications, particularly in extremal graph theory. A regularity lemma for 3-uniform hypergraphs developed by Frankl and Rödl [8] allows some of the Szemerédi Regularity Lemma graph applications to be extended to hypergraphs. An important development regarding Szemerédi's Lemma showed the equivalence between the property of ε-regularity of a bipartite graph G and an easily verifiable property concerning the neighborhoods of its vertices (Alon et al. [1]; cf. [6]). This characterization of ε-regularity led to an algorithmic version of Szemerédi's lemma [1]. Similar problems were also considered for hypergraphs. In [2], [9], [13], and [18], various descriptions of quasi-randomness of k-uniform hypergraphs were given. As in [1], the goal of this paper is to find easily verifiable conditions for the hypergraph regularity provided by [8]. The hypergraph regularity of [8] renders quasi-random "blocks of hyperedges" which are very sparse. This situation leads to technical difficulties in its application. Moreover, as we show in this paper, some easily verifiable conditions analogous to those considered in [2] and [18] fail to be true in the setting of [8]. However, we are able to find some necessary and sufficient conditions for this hypergraph regularity. These conditions enable us to design an algorithmic version of a hypergraph regularity lemma in [8]. This algorithmic version is presented by the authors in [5].