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...tices has DISCd(δ) for d, δ > 0, if e(U) = d (|U | k )± δnk for all U ⊆ V (Hn) , where by x = y ± z we mean that x lies in the interval [y − z, y + z]. Hypergraphs with property DISCd were studied in =-=[2, 3, 18]-=- and a straightforward generalisation of Szemerédi’s regularity lemma for this concept was observed to hold in [4, 11, 30] (see Theorem 23 below). We will suggest extensions of properties P1, P2, P6,...

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...used the so-called weak hypergraph regularity lemma (see e.g. [5, 11, 32]), which is a straightforward generalization of Szemerédi’s regularity lemma for graphs. This version has recently been proved =-=[15]-=- to be compatible (i.e., admits a counting lemma) with linear hypergraphs, i.e., those hypergraphs whose hyperedges intersect pairwise in at most one vertex. In this setting the notion of the cluster ...

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...diately after proving their graph theorem, Chung and Graham [3, 4, 5, 6, 7] began investigating a k-uniform hypergraph generalization. Since then, many authors have studied hypergraph quasirandomness =-=[2, 9, 10, 13, 14, 15, 16, 17, 19, 22]-=-. One important k-uniform hypergraph quasirandom property is Disc, which states that all sufficiently large vertex sets have roughly the same edge density as the entire hypergraph. Kohayakawa, Nagle, ...

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...quivalent to counting some class of hypergraphs or counting a single substructure. This is related to the Weak Hypergraph Regularity Lemma [10, 20, 39]. Recently, Kohayakawa, Nagle, Rödl, and Schacht =-=[28]-=- answered this question by showing that Disc is equivalent to counting the family of linear hypergraphs, where a hypergraph H is linear if every pair of distinct edges share at most one vertex. Buildi...

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... set V ∗ = [t] = {1, . . . , t} and edge set E∗ , where for 1 ≤ i < j < k ≤ t it is {i, j, k} ∈ E∗ if and only if the triple (Vi, Vj, Vk) is ε-regular and the density satisfies dH(Vi, Vj, Vk) ≥ γ. In =-=[12]-=- a counting lemma for linear hypergraphs in the context of the weak hypergraph regularity lemma was proved, where a hypergraph is said to be linear if no two of its hyperedges intersect in more than o...

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...) of hyperedges, where for 1 ≤ i1 < · · · < ik ≤ t it is {i1, . . . , ik} ∈ E(H(γ)) if and only if the k-tuple (Vi1 , . . . , Vik) is ε-regular and its density satisfies dH(Vi1 , . . . , Vik) ≥ γ. In =-=[15]-=-, in the context of the weak hypergraph regularity lemma, a counting lemma for linear hypergraphs was proved, from which one obtains the following. Lemma 5. Let F be a fixed k-uniform, linear hypergra...

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... regularity lemma, which is almost identical to Szemerédi’s regularity lemma. They then proved an embedding lemma for linear hypergraphs essentially following ideas from Kohayakawa-Nagle-Rödl-Schacht =-=[13]-=-. It is well-known that such an embedding lemma fails to hold for non-linear 3-graphs unless one uses the (strong) 3-graph regularity lemma, and operating in this environment is more complicated. In t...