Is induced subgraph isomorphism easy on an infinite subclass? Is there a sequence of undirected graphs $\{C_n\}_{n\in \mathbb N}$, 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.