The Hyperclique Hypothesis is False

Abstract

The Hyperclique Hypothesis postulates that, for all constants $k > h \geq 3$, detecting a $k$-clique in an $h$-uniform hypergraph with $n$ vertices requires time $n^{k-o(1)}$. This hypothesis is false: there is an algebraic algorithm that detects (and even counts) $k$-cliques in $h$-uniform hypergraphs in time $O(n^{(1-ε_h)k})$. Via known reductions, this also refutes the recently proposed $k$-AP Hypothesis. As a consequence, many problems with conditional lower bounds based on these two hypotheses may admit faster algorithms after all. This result is almost entirely due to AI.