We present an algorithm that solves the $k$‑Clique problem for any sufficiently large $k$ and any $n$‑vertex graph in $n^{(\frac{\omega}{3}-c)k}$ time, where $\omega\ge 2$ is the matrix multiplication exponent and $c>0$ is a constant independent of both $\omega$ and $k$. This result directly refutes the long‑standing $k$‑Clique conjecture, regardless of the matrix multiplication constant, even when $\omega=2$. Moreover, the algorithm can count weighted $k$‑Cliques.
The key idea is to reduce Clique detection to several matrix products, apply a block‑wise fast matrix multiplication technique, and introduce a constant factor $c$ at each level, thereby shaving off a $\frac{\omega}{3}$ term in the exponent. This approach theoretically breaks the previous lower‑bound barrier tied to $\omega$.
Review: The breakthrough suggests that, even under the assumption that matrix multiplication remains the main bottleneck, the lower bounds for $k$‑Clique must be revisited.