Parameterized complexity of the maximum independent set problem and the speed of hereditary properties

For a hereditary class X, the number Xn of n-vertex graphs in X (also known as the speed of X) satisfies where k(X) is a natural number called the index of the class. Each class X of index k>1 can be approximated by a minimal class of the same index. In this paper, we use Ramsey theory to show that the maximum independent set problem is fixed-parameter tractable in all minimal classes of index k for all values of k.