The following exercise is difficult for me:Show that for each $k \in \mathbb{N}$ the question of existence of a $k$-clique within a graph lies in $\text{L}$.Hint: A $k$-clique denotes $k$ verteces within a graph that are all connected with each other.Annotation: if the question also considers $k$ as parameter, then the problem is $\text{NP}$-complete.So $\text{L}$ is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a logarithmic amount of memory space. Now I'm wondering if this exercise isn't a catchy question, since I've found this paper and it says We give an algorithm for $k$-clique that runs in (...) $O(n^{\varepsilon})$ space, for all $\varepsilon > 0$, on graphs with $n$ nodes?Furthermore, I also don't understand the annotation, because how should it be possible to not consider $k$ for the $k$-clique problem?
Show that k-clique lies in L
complexity theory;space complexity