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

For a hereditary class X, the number X n of n-vertex graphs in X (also known as the speed of X) satisfies lim n → ∞ log 2 X n ( n 2 ) = 1 − 1 k ( X ) 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.