Multigraphs (Only) Satisfy a Weak Triangle Removal Lemma

Abstract

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.