Researchers have presented an algorithm that breaks the widely‑held k‑Clique conjecture for every large enough clique size k and any n‑vertex graph. The time needed is n^{(ω/3−c)k}, where ω is the matrix‑multiplication exponent (any value ≥2) and c is a positive constant that does not depend on ω or k. This result disproves the conjecture for all k and for every matrix‑multiplication exponent, even if ω=2. The same technique also counts weighted k‑Cliques: it sums the weights of all k‑cliques in a weighted graph.

For cliques whose size is a multiple of 9 the algorithm achieves a further saving. If k=9t, the running time becomes n^{(ω/3−α_{36}/9)k+o(1)}, an improvement over the previous n^{ωk/3} bound. The authors also show that any weighted pattern graph on nine vertices can be counted in time n^{3ω−α_{36}+o(1)}, better than the trivial n^{3ω} time for all such patterns.

The authors do not give explicit numerical values for the constants c and α_{36}. Their analysis purposely omits optimization so the precise exponent improvement cannot be computed. The method also relies on an unseen universal constant δ from a rank‑or‑swap decomposition, leaving the exact size of the speed‑up unknown.

These findings illustrate how new algebraic techniques can overturn longstanding assumptions in graph theory, although the practical impact awaits a clearer understanding of the hidden constants.