Regular clique counting lemma ($\#K_r\ge \delta n^r$)

For fixed $r$ and positive density threshold $\lambda$, sufficiently regular pairwise dense bipartite graphs between $r$ disjoint equal-sized vertex classes contain at least $\delta n^r$ transversal cliques, for some $\delta >0$. A greedy embedding maintains common neighbourhoods and discards the small set of vertices with atypical degrees into them. The constants depend only on $r,\lambda$.

If three disjoint vertex classes have size $L$, and their three pairs are $\epsilon$-regular pairs of vertex sets of edge density of a bipartite graph at least $d$, where $0<\epsilon \le d/2$ and $\epsilon <1/2$, they contain at least the displayed number of transversal triangles in a graph. All but $2\epsilon L$ vertices in the first class have at least $(d-\epsilon )L$ neighbours in both other classes. Regularity between those two neighbour sets supplies the third edge. This strengthens a mere triangle embedding lemma for regular pairs to a cubic lower count.