An efficient asymmetric removal lemma and its limitations

The triangle removal states that if G contains edge-disjoint triangles, then G contains triangles. Unfortunately, there are no sensible bounds on the order of growth of , and at any rate, it is known that is not polynomial in . Csaba recently obtained an asymmetric variant of the triangle removal, stating that if G contains edge-disjoint triangles, then G contains copies of . 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 to the optimal number . • We say that H is -abundant if every graph containing edge-disjoint triangles has copies of H. It is easy to see that a -abundant graph must be triangle-free and tripartite. Given our first result, it is natural to ask if all triangle-free tripartite graphs are -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.