It's well known that problems like $3$-sat and $4$-sat and probably $k$-sat for $k\geq 5$ are NP-hard problems but what happens for example if i was to consider something like $\lceil \mathrm{log}(n) \rceil$-sat where the number of literals in each clause grows as a function of $n$ (number of variables), is it difficult to prove hardness of such a result?
Another example might be deciding if a graph has a clique of size $k(n) \rightarrow \infty$ as $n \rightarrow \infty$ where $n$ is the number of vertices. Is it difficult to prove hardness of problems of this form where input parameters are non-constant? Could anyone provide links where results like these are proved? Any references are highly appreciated.
NP-hardness of your $\lceil \log n\rceil$-SAT follows immediately by the fact that $\lceil \log n\rceil \ge 3$ whenever $n\ge 5$ -- and instances with less than $5$ variables are easy anyway.
(Note that you can convert a $k$-SAT instance to a $K$-SAT for $K>k$ simply by adding redundant literals to each clause.)
$k(n)$-CLIQUE is hard for at least some choices of $k$. For example, if you have an efficient $\lfloor n/2\rfloor$-CLIQUE solver and want to find an $m$-clique in a graph with $n$ vertices, then
However, for other functions $k$, such as $k=\lfloor \log n\rfloor$ it is not obvious that $k(n)$-clique is either NP-hard or in P. The problem with generalizing the above reduction is that if you want to find a medium-sized clique in a large graph, it may take superpolynomially many added nodes to make $n$ large enough that $k(n)$ matches your desired clique size.