[Solved]: Is induced subgraph isomorphism easy on an infinite subclass?

Problem Detail: Is there a sequence of undirected graphs ${C_n}_{nin mathbb N}$, where each $C_n$ has exactly $n$ vertices and the problem

Given $n$ and a graph $G$, is $C_n$ an induced subgraph of $G$?

is known to be in class $mathsf{P}$?

This question has been answered on cstheory. Digest: Chen,Thurley and Weyer (2008) prove that this problem is $W[1]$-hard for every infinite class of graphs.