Applications of the regularity lemma for uniform hypergraphs
Random Structures & Algorithms
The counting lemma for regular k-uniform hypergraphs
Random Structures & Algorithms
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A bipartite graph G = (V1 ∪ V2, E) is (&dgr;, d)-regular if $$|d-d(V^{\prime}_{1}, V^{\prime}_{2})| G = (V1 ∪ V2 ∪ V3, E) is a 3-partite graph whose restrictions on V1 ∪ V2, V1 ∪ V3, V2 ∪ V3 are (&dgr;, d)-regular, then G contains $(d^{3} \pm f(\delta))|V_{1}||V_{2}||V_{3}|$ copies of K3. This fact and its various extensions are the key ingredients in most applications of Szemerédi's Regularity Lemma. To derive a similar results for r-uniform hypergraphs, r 2, is a harder problem. In 1994, Frankl and Rödl developed a regularity lemma and counting argument for 3-uniform hypergraphs. In this paper, we exploit their approach to develop a counting argument for 4-uniform hypergraphs. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005