Removal lemmas and approximate homomorphisms

We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each ϵ>0 there exists M such that every triangle-free graph G has an ϵ -approximate homomorphism to a triangle-free graph F on at most M vertices (here an ϵ -approximate homomorphism is a map V(G)→V(F) where all but at most ϵ|V(G)|2 edges of G are mapped to edges of F). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in ϵ−1 . We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal