Note on the 3-graph counting lemma

Szemeredi's regularity lemma proved to be a powerful tool in extremal graph theory. Many of its applications are based on the so-called counting lemma: if G is a k-partite graph with k-partition V"1@?...@?V"k, |V"1|=...=|V"k|=n, where all induced bipartite graphs G[V"i,V"j] are (d,@e)-regular, then the number of k-cliques K"k in G is d^k^2n^k(1+/-o(1)). Frankl and Rodl extended Szemeredi's regularity lemma to 3-graphs and Nagle and Rodl established an accompanying 3-graph counting lemma analogous to the graph counting lemma above. In this paper, we provide a new proof of the 3-graph counting lemma.

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