Before revision: We give an algorithm that takes as input a graph G with weights on the vertices and outputs a maximum weight independent set of G . If G  does not contain any cycle on k or more vertices as an induced subgraph (G is a C _{ \geq k }-free graph ), the algorithm runs in time n^{O( k^3 \log ^5 n) , and therefore for fixed k is a quasi-polynomial time algorithm.
After revision: We give an algorithm that takes as input a graph G with weights on the vertices and outputs a maximum weight independent set of G . If G does not contain any induced cycle on more than~t vertices. We prove the following statement: for every pair of integers d and t and a CMSO_2 statement~\phi, there exists an algorithm that, given an n-vertex C _{ \geq k }-free graph ), the algorithm runs in time n^{O( k^3 \log ^5 n) , and therefore for fixed k is a quasi-polynomial time algorithm.