Multigraphs (only) satisfy a weak triangle removal lemma (Q1028803)

Summary: The triangle removal lemma states that a simple graph with \(o(n^3)\) triangles can be made triangle-free by removing \(o(n^2)\) edges. It is natural to ask if this widely used result can be extended to multi-graphs. In this short paper we rule out the possibility of such an extension by showing that there are multi-graphs with only \(n^{2+o(1)}\) triangles that are still far from being triangle-free. On the other hand, we show that for some slowly growing function \(g(n)=\omega(1)\), if a multi-graph has only \(g(n)n^2\) triangles then it must be close to being triangle-free. The proof relies on variants of the Ruzsa-Szemerédi theorem.