  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).
