Show that k-clique lies in L

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?

If you consider $k$ a constant, then k-clique can be solved be enumerating all $O(n^k)$ sets of vertices and checking that there are edges present in polynomial time (since the exponent is constant).

Enumerating k-Tuples of numbers up to n can be done in $k\log n$ space, you only have to figure out how to check for the presence of an edge between two vertices without using much space. Depending on how much details you have to give this is either obvious or very tedious to write down.

StackExchange Computer Science Q#7039, answer score: 5