An efficient asymmetric removal lemma and its limitations

The triangle removal states that if G contains ϵn² edge-disjoint triangles, then G contains δ (ϵ)n³ triangles. Unfortunately, there are no sensible bounds on the order of growth of δ(ϵ), and at any rate, it is known that δ(ϵ) is not polynomial in ϵ. Csaba recently obtained an asymmetric variant of the triangle removal, stating that if G contains ϵn² edge-disjoint triangles, then G contains 2^-poly(1-ϵ) · n⁵ copies of n⁵. To this end, he devised a new variant of Szemerédi's regularity lemma. We obtain the following results: • We first give a regularity-free proof of Csaba's theorem, which improves the number of copies of C₅ to the optimal number poly (ϵ) · n⁵. • We say that H is K₃-abundant if every graph containing ϵn² edge-disjoint triangles has poly (ϵ) · n^|V(H)| copies of H. It is easy to see that a K₃-abundant graph must be triangle-free and tripartite. Given our first result, it is natural to ask if all triangle-free tripartite graphs are K₃-abundant. Our second result is that assuming a well-known conjecture of Ruzsa in additive number theory, the answer to this question is negative. Our proofs use a mix of combinatorial, number-theoretic, probabilistic and Ramsey-type arguments.