Lincoln–Vassilevska-Williams–Williams hyperclique conjecture

Let $k>r\geq 3$ and let an $r$-uniform hypergraph be a hypergraph whose edges have size $r$. A $k$-hyperclique is a set of $k$ vertices such that every $r$-subset is an edge. The Lincoln–Vassilevska-Williams–Williams hyperclique conjecture states that, for every $k>r\geq 3$ and every $\varepsilon>0$, $k$-hyperclique detection in $r$-uniform hypergraphs cannot be solved in

$$O(n^{k-\varepsilon})$$

time. This conjecture underlies conditional lower bounds in fine-grained complexity, and the source notes that several breakthroughs would follow if it were false.