Note on the 3-graph counting lemma Original Research Article

Szemerédiʹs 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 image is a image-partite graph with image-partition image, image, where all induced bipartite graphs image are image-regular, then the number of image-cliques image in image is image. Frankl and Rödl extended Szemerédiʹs 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.