Let $\phi$ be negative monotone 2-CNF on $n$ variables and $n^{3/2}$ clauses.
What is the complexity of finding satisfying assignment with maximum number of ones $k$?
Alternatively let $G$ be a graph of order $n$ and $n^{3/2}$ edges. $G$ is dense.
What is the complexity of finding $k$-independent set of $G$?
Each variables $x_i$ is in about $\sqrt{n}$ clauses (alternatively the degrees of vertices of $G$ are about $\sqrt{n}$).
Can we get fixed parameter tractable algorithm with parameter $k$?
Can we increase the exponent $3/2$ to get polynomial solutions?
If necessary assume $n$ is square to get rid of the fractions.
Added later
Third question:
Let $\frac12 \le C < n$ and $d=Cn$. Let $G$ be $d$-regular graph. It has $C n^2/2$ edges and each vertex has $d$ neighbors.
In CNF notation there are $C n^2/2$ clauses and each variable is in $d$ clauses.
Can we find MIS in $G$ in subexponential or polynomial time?
More details about this construction are on Mathoverlfow