Before revision: In a recent  work, Gartland and Lokshtanov [FOCS 2020] gave a quasi-polynomial-time algorithm for Maximum Weight Independent Set in P_t-free graphs, that is, graphs excluding a path on t vertices as an induced subgraph.
After revision: In a recent breakthrough work, Gartland and Lokshtanov [FOCS 2020] gave a quasi-polynomial-time algorithm for Maximum Weight Independent Set in P_t-free graphs, that is, graphs excluding a path on t vertices as an induced subgraph.