Before revision:  A subexponetial time algorithm for C_{\geq 5}-free graphs (also known as long-hole-free graphs) was found in 2019 }  [ Chudnovsky et al., Arxiv'19 ] followed by a polynomial time algorithm for C_{\geq 5 -free graphs in 2020 [ Abrishami et al. , Arxiv'20 ] . For k > 5 only a quasi-polynomial time approximation scheme Chudnovsky et al., SODA'20 was known. Our work is the first to exhibit conclusive evidence that Independent Set on C_{\geq k{\sc{Maximum Weight Independent Set}}  -free graphs isnot NP-complete for any integer k.
After revision:  A subexponetial time algorithm for C_{\geq 5}-free graphs (also known as long-hole-free graphs) was found in 2019 } [ Chudnovsky et al., Arxiv'19 ] followed by a polynomial time algorithm for C_{\geq 5 -free graphs in 2020 [ Abrishami et al. , Arxiv'20 ] . For k > 5 only a quasi-polynomial time approximation scheme Chudnovsky et al., SODA'20 on the{\sc{Maximum Weight Independent Set}} -free graphs isnot NP-complete for any integer k.