Multigraphs (only) satisfy a weak triangle removal lemma

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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 (or equivalently, weighted 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 g(n)=ω(1), if a multi-graph (or weighted 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.