Clique problem on fixed graphs
We did some research on the problem of proving in tree-like Resolution whether a fixed graph $G$ has a clique of size $k$ (where $k$ is usually small). In particular we discovered that refutations of size $n^{\Omega(k)}$ are needed for a large class of graphs. 

You can find the paper Parameterized Complexity of DPLL Search Procedures at this link.