Complexity of MAX-ONEs Monotone 2-SAT with $n^{3/2}$ or $C n^2$ clauses?
In both questions the answer is the same for any exponent $1 &lt; \alpha &lt; 2$.
For the first question, we can use a degeneracy based algorithm to find an independent set of size $\Omega(\frac{n}{n^{\alpha-1}}) = \Omega(n^{2-\alpha})$ in polynomial time. Therefore we get an FPT algorithm for every constant $\alpha$ simply by outputting YES if $k &lt; c n^{2-\alpha}$ for some constant $c$, and otherwise solving the problem with bruteforce in $2^n = 2^{O(k^{1/(2-\alpha)})}$ time.
For the second question, we can prove that the problem is NP-hard for all $1 &lt; \alpha &lt; 2$ by reducing from maximum independent set problem in 3-regular graphs, which is NP-hard [1].
For some constant $\beta$ we make $n^\beta$ copies of each vertex, making the set of copies a clique for each vertex. Now the new graph has exactly the same maximum independent set as the old graph. It is $4 n^\beta - 1$ regular and has $n^{\beta+1}$ vertices, so we can choose $\beta \approx \frac{\alpha-1}{2-\alpha}$ to make it work for any $1 &lt; \alpha &lt; 2$.
[1] Garey, Michael R., David S. Johnson, and Larry Stockmeyer. &quot;Some simplified NP-complete problems.&quot; STOC 1974.