Note on the 3-graph counting lemma

Szemerédi'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 GG is a kk-partite graph with kk-partition V1∪⋯∪VkV1∪⋯∪Vk, |V1|=⋯=|Vk|=n|V1|=⋯=|Vk|=n, where all induced bipartite graphs G[Vi,Vj]G[Vi,Vj] are (d,ε)(d,ε)-regular, then the number of kk-cliques KkKk in GG is dk2nk(1±o(1)). Frankl and Rödl extended Szemerédi's regularity lemma to 3-graphs and Nagle and Rödl 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.