article published yes Lior Gishboliner author Asaf Shapira author Yuval Wigderson author 2d0023a0-1567-11f0-833d-d5c1e476d4b5 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. Cambridge University Press2025 eng 2050-5094 2301.0769310.1017/fms.2024.68 13 yes Gishboliner, Lior, Asaf Shapira, and Yuval Wigderson. “An Efficient Asymmetric Removal Lemma and Its Limitations.” <i>Forum of Mathematics, Sigma</i>. Cambridge University Press, 2025. <a href="https://doi.org/10.1017/fms.2024.68">https://doi.org/10.1017/fms.2024.68</a>. L. Gishboliner, A. Shapira, Y. Wigderson, Forum of Mathematics, Sigma 13 (2025). Gishboliner, Lior, et al. “An Efficient Asymmetric Removal Lemma and Its Limitations.” <i>Forum of Mathematics, Sigma</i>, vol. 13, e38, Cambridge University Press, 2025, doi:<a href="https://doi.org/10.1017/fms.2024.68">10.1017/fms.2024.68</a>. L. Gishboliner, A. Shapira, and Y. Wigderson, “An efficient asymmetric removal lemma and its limitations,” <i>Forum of Mathematics, Sigma</i>, vol. 13. Cambridge University Press, 2025. Gishboliner, L., Shapira, A., &#38; Wigderson, Y. (2025). An efficient asymmetric removal lemma and its limitations. <i>Forum of Mathematics, Sigma</i>. Cambridge University Press. <a href="https://doi.org/10.1017/fms.2024.68">https://doi.org/10.1017/fms.2024.68</a> Gishboliner L, Shapira A, Wigderson Y. An efficient asymmetric removal lemma and its limitations. <i>Forum of Mathematics, Sigma</i>. 2025;13. doi:<a href="https://doi.org/10.1017/fms.2024.68">10.1017/fms.2024.68</a> Gishboliner L, Shapira A, Wigderson Y. 2025. An efficient asymmetric removal lemma and its limitations. Forum of Mathematics, Sigma. 13, e38. 221582026-06-29T10:51:07Z2026-07-08T10:31:22Z