We revisit the algorithmic problem of finding a triangle in a graph: We give a randomized combinatorial algorithm for triangle detection in a given $n$-vertex graph with $m$ edges running in $O(n^{7/3})$ time, or alternatively in $O(m^{4/3})$ time. This may come as a surprise since it invalidates several conjectures in the literature. In particular, - the $O(n^{7/3})$ runtime surpasses the long-standing fastest algorithm for triangle detection based on matrix multiplication running in $O(n^\omega) = O(n^{2.372})$ time, due to Itai and Rodeh (1978). - the $O(m^{4/3})$ runtime surpasses the long-standing fastest algorithm for triangle detection in sparse graphs based on matrix multiplication running in $O(m^{2\omega/(\omega+1)})= O(m^{1.407})$ time due to Alon, Yuster, and Zwick (1997). - the $O(n^{7/3})$ time algorithm for triangle detection leads to a $O(n^{25/9} \log{n})$ time combinatorial algorithm for $n \times n$ Boolean matrix multiplication, by a reduction of V. V. Williams and R.~R.~Williams (2018).This invalidates a conjecture of A.~Abboud and V. V. Williams (FOCS 2014). - the $O(m^{4/3})$ runtime invalidates a conjecture of A.~Abboud and V. V. Williams (FOCS 2014) that any combinatorial algorithm for triangle detection requires $m^{3/2 -o(1)}$ time. - as a direct application of the triangle detection algorithm, we obtain a faster exact algorithm for the $k$-clique problem, surpassing an almost $40$ years old algorithm of Ne{\v{s}}et{\v{r}}il and Poljak (1985). This result strongly disproves the combinatorial $k$-clique conjecture. - as another direct application of the triangle detection algorithm, we obtain a faster exact algorithm for the \textsc{Max-Cut} problem, surpassing an almost $20$ years old algorithm of R.~R.~Williams (2005).

翻译：我们重新审视图中寻找三角形的算法问题：对于给定一个包含$n$个顶点、$m$条边的图，我们提出了一种随机组合算法，该算法可在$O(n^{7/3})$时间或$O(m^{4/3})$时间内完成三角形检测。这一结果可能出人意料，因为它推翻了文献中的若干猜想。具体而言： - $O(n^{7/3})$的运行时间超越了由Itai和Rodeh（1978年）提出的、基于矩阵乘法的长期最快三角形检测算法（运行时间为$O(n^\omega) = O(n^{2.372})$）。 - $O(m^{4/3})$的运行时间超越了由Alon、Yuster和Zwick（1997年）提出的、针对稀疏图的基于矩阵乘法的长期最快三角形检测算法（运行时间为$O(m^{2\omega/(\omega+1)})= O(m^{1.407})$）。 - 针对三角形检测的$O(n^{7/3})$时间算法，通过V. V. Williams和R. R. Williams（2018年）的归约，可导出一个用于$n \times n$布尔矩阵乘法的$O(n^{25/9} \log{n})$时间组合算法。这推翻了A. Abboud和V. V. Williams（FOCS 2014年）的一个猜想。 - $O(m^{4/3})$的运行时间推翻了A. Abboud和V. V. Williams（FOCS 2014年）的猜想，即任何用于三角形检测的组合算法都需要$m^{3/2 -o(1)}$时间。 - 作为三角形检测算法的直接应用，我们获得了用于$k$-团问题的更快精确算法，超越了Ne{\v{s}}et{\v{r}}il和Poljak（1985年）近40年前的算法。这一结果有力地否定了组合$k$-团猜想。 - 作为三角形检测算法的另一直接应用，我们获得了用于\textsc{Max-Cut}问题的更快精确算法，超越了R. R. Williams（2005年）近20年前的算法。