Consider a graph with $n$ vertices and maximum degree $\Delta$. I would like to find if the graph has any $s$ cliques, where $s \leq \Delta$ and both of them are small compared to $n$. I only need to find a single such clique (or certify that none exist)
There is a straightforward way to do this: for each vertex $v$, test all $s$-subsets of the neighbors of $v$. The work is thus $\approx n \binom{\Delta}{s-1}$.
Are there any more efficient algorithms than this? Even achieving an exponential speed-up would be good?