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 $n>2$:

At first, $c(n)\leq n$. Indeed, no two edges may belong to the same clique, else the cycle is not induced. Example of induced cycle of length $n$: all edges belong to different cliques, also, there are common vertices for couples of non-adjacent cliques.