Complexity of k-clique for hypergraphs
It is not known if there is an $\varepsilon &gt; 0$, $c &gt; 2$, and $k &gt; c$ such that $(c,k)$ hyperclique is in $n^{k-\varepsilon}$ time. Note that the case of $k \leq c$ is trivial. For years I have communicated this problem to many people, and taught it in cs266 at Stanford, due to its connection to solving $k$-Sat. (Several open problem sessions at workshops probably recorded this.)  Here are a few things I know:

I proved several years ago that solving $4-cycle$ on $n$ node graphs in $n^{2-\varepsilon}$ time implies $(3,4)$ hyperclique in $n^{4-\varepsilon}$ time. Haven't published it. 

UPDATE (Aug 2019) the aforementioned result and some generalizations now appear in the paper

Andrea Lincoln, Virginia Vassilevska Williams, R. Ryan Williams:
Tight Hardness for Shortest Cycles and Paths in Sparse Graphs. SODA 2018: 1236-1252

If you can solve $(3,4)$ hyperclique as indicated above, then Max-3-Sat can be solved in strictly less than $2^n$ time. Similarly, solving $(k,k+1)$ hyperclique would yield a faster $k$-Sat algorithm. So if you believe Strong ETH then there is an obvious limit here. The reduction is a natural generalization of the reduction from Max-2-Sat to triangle finding ($(2,3)$ clique) from ICALP'04 and my PhD thesis. 

You can solve $(c,k)$ hyperclique in $n^k/(\log n)^{\Omega(k)}$ time by generalizing the paper Efficient Algorithms for Clique Problems.