For any set $X$ we set $[X]^2 = \big\{\{a,b\}: a, b\in X\text{ and } a\neq b\big\}$.


We say a simple undirected graph $G=(V,E)$ is an $n$-*clique graph* if there are $S\_1,\ldots,S\_n\subseteq V$ such that


1. $|S\_k| = n$ for all $k=1,\ldots, n$;
2. $V = \bigcup\_{k=1}^n S\_k$;
3. $i\neq k \in \{1,\ldots,n\}$ implies $|S\_i \cap S\_k| = 1$.
4. $E = \bigcup\_{k=1}^n [S\_k]^2$ (that is, all the $S\_k$ are complete, and there are no edges between different $S\_k$.)


Let $c(n)$ be the maximum length of an [induced cycle](https://en.wikipedia.org/wiki/Induced_path) that any $n$-clique graph $G$ can have. Is there an explicit formula for $c(n)$, and if not, what is $\lim\_{n\to\infty}\frac{c(n)}{n}$?



Maximal induced cycles on $n$-clique graphs
I am not sure that I understand all conditions correctly, since the question looks too straightforward, but if yes, then there are no cycles for $n\leq 2$ and for $n>2$ I claim that $c(n)=n$.


At first, $c(n)\leq n$. Indeed, no two edges of our cycle $x\_1\dots x\_{c(n)}x\_1$ may belong to the same clique (if $c(n)\geq n+1\geq 4$), else the cycle is not induced. 


Example of induced cycle $1\dots n$ of length $n$: $S\_i$ contains vertices $i,i+1$ (modulo $n$, of course), and $n-2$ vertices $v\_{i,j}$, $j\in \{1,\dots,n\}\setminus \{i+1,i-1\}$.


