Let N be the number of triangles in an Erdős–Rényi graph \({\mathcal {G}}(n,p)\) on n vertices with edge density \(p=d/n,\) where \(d>0\) is a fixed constant. It is well known that N weakly converges to the Poisson distribution with mean \({d^3}/{6}\) as \(n\rightarrow \infty \). We address the upper tail problem for N, namely, we investigate how fast k must grow, so that \({\mathbb {P}}(N\ge k)\) is not well approximated anymore by the tail of the corresponding Poisson variable. Proving that the tail exhibits a sharp phase transition, we essentially show that the upper tail is governed by Poisson behavior only when \(k^{1/3} \log k< (\frac{3}{\sqrt{2}} - {o(1)})^{2/3} \log n\) (sub-critical regime) as well as pin down the tail behavior when \(k^{1/3} \log k> (\frac{3}{\sqrt{2}} + {o(1)})^{2/3} \log n\) (super-critical regime). We further prove a structure theorem, showing that the sub-critical upper tail behavior is dictated by the appearance of almost k vertex-disjoint triangles whereas in the supercritical regime, the excess triangles arise from a clique like structure of size approximately \((6k)^{1/3}\). This settles the long-standing upper-tail problem in this case, answering a question of Aldous, complementing a long sequence of works, spanning multiple decades and culminating in Harel et al. (Duke Math J 171(10):2089–2192, 2022), which analyzed the problem only in the regime \(p\gg \frac{1}{n}.\) The proofs rely on several novel graph theoretical results which could have other applications.

令N为 Erdős–Rényi 图\({\mathcal {G}}(n,p)\)中n 个顶点上边密度为\(p=d/n,\)的三角形数量，其中\(d> 0\)是固定常数。众所周知，N弱收敛于泊松分布，均值\({d^3}/{6}\)为\(n\rightarrow \infty \)。我们解决N的上尾问题，即我们研究k必须以多快的速度增长，以便\({\mathbb {P}}(N\ge k)\)不再被相应泊松的尾部很好地近似多变的。证明尾部表现出急剧的相变，我们本质上表明，仅当\(k^{1/3} \log k< (\frac{3}{\sqrt{2}} - {o(1)})^{2/3} \log n\)（亚临界状态）以及确定\(k^{1/3} \log k> (\frac {3}{\sqrt{2}} + {o(1)})^{2/3} \log n\)（超临界状态）。我们进一步证明了一个结构定理，表明亚临界上尾行为是由几乎k 个顶点不相交三角形的出现决定的，而在超临界状态下，多余的三角形来自尺寸约为\((6k )^{1/3}\) .这解决了本案中长期存在的上尾问题，回答了奥尔德斯的问题，补充了一系列跨越数十年的作品，并在哈雷尔等人的著作中达到顶峰。 （Duke Math J 171(10):2089–2192, 2022），仅在\(p\gg \frac{1}{n}.\)体系中分析了该问题。证明依赖于几个新颖的图理论结果，这些结果可以有其他应用。