We prove that any n-vertex graph whose complement is triangle-free contains n2/12 – o(n2) edge-disjoint triangles. This is tight for the disjoint union of two cliques of order n/2. We also prove a corresponding stability theorem, that all large graphs attaining the above bound are close to being bipartite. Our results answer a question of Alon and Linial, and make progress on a conjecture of Erdős.

中文翻译：

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无协三角形图中许多不相交的三角形

我们证明任何n- 补码为无三角形的顶点图包含n2/12 –○(n2) 边不相交的三角形。这对于两个秩序集团的不相交联合来说是紧密的n/2。我们还证明了一个相应的稳定性定理，即所有达到上述界限的大图都接近于二分图。我们的结果回答了 Alon 和 Linial 的问题，并在 Erdős 的猜想上取得了进展。