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{\bf Asaf Shapira and Raphael Yuster}
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{\bf Multigraphs (Only) Satisfy a Weak Triangle Removal Lemma}
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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-Szemer\'edi theorem.



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