Let a number $k$ be given. The $k$-clique problem is as follows.
Given a graph $G$, does there exist a subset $S$ of $k$ vertices so that any two vertices of $S$ are adjacent?
Let numbers $c$ and $k$ be given. The $(c,k)$-hyperclique problem is as follows.
Given a $c$-uniform hypergraph $H$, does there exist a set $S$ of $k$ vertices so that any subset of $c$ vertices from $S$ forms a hyperedge.
(1) What is the best known algorithm for solving $(c,k)$-hyperclique?
(2) What is its time complexity?
(3) Is there any connection between $(c,k)$-hyperclique and matrix multiplication?
For all I know, this might be a well studied problem. Any references that investigate this problem are greatly appreciated.