Multigraphs (Only) Satisfy a Weak Triangle Removal Lemma

The triangle removal lemma states that a simple graph with o(n3) triangles can be made triangle-free by removing o(n2) 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 n2+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) = ω(1), if a multi-graph has only g(n)n2 triangles then it must be close to being triangle-free. The proof relies on variants of the Ruzsa-Szemerédi theorem [15].