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.
After revision:  A subexponetial time algorithm for C_{\geq 5}-free graphs (also known as long-hole-free graphs) was found in 2019 } a maximum-weight vertex subset S such that G [ 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.