Counting subgraphs in quasi-random 4-uniform hypergraphs

A bipartite graph G = (V1 ∪ V2, E) is (δ, d)-regular if ̨̨ d− d(V ′ 1 , V ′ 2 ) ̨̨ < δ whenever V ′ i ⊂ Vi, |V ′ i | ≥ δ|Vi|, i = 1, 2.Here, d(V ′ 1 , V ′ 2 ) = e(V ′ 1 , V ′ 2 )/|V ′ 1 ||V ′ 2 | stands for the density of the pair (V ′ 1 , V ′ 2 ). An easy counting argument shows that if G = (V1 ∪ V2 ∪ V3, E) is a 3-partite graph whose restrictions on V1 ∪V2, V1 ∪V3, V2 ∪V3 are (δ, d)regular, then G contains (d ± f(δ))|V1||V2||V3| 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.