We consider the parameterized complexity of the following problem under the framework introduced by Downey and Fellows: Given a graph $G$, an integer parameter $k$ and a nontrivial hereditary property $\Phi$, are there $k$ vertices of $G$ that induce a subgraph with property $\Phi$? This problem has been proved $\mathsf{NP}$-hard by Lewis and Yannakakis. We show that if $\Phi$ includes all trivial graphs but not all complete graphs or vice versa, then the problem is complete for the parameterized class $\mathsf{W}[1]$ and is fixed parameter tractable otherwise. Our proofs of both the tractability and hardness involve nontrivial use of the theory of Ramsey numbers.
(Based on a paper by Subhash Khota, Venkatesh Raman in Theor. Comput. Sci. 2002.)