  We prove that a number of computational problems that ask for the largest
sparse induced subgraph satisfying some property definable in CMSO2 logic, most
notably Feedback Vertex Set, are polynomial-time solvable in the class of
$P_6$-free graphs. This generalizes the work of Grzesik, Klimo\v{s}ov\'{a},
Pilipczuk, and Pilipczuk on the Maximum Weight Independent Set problem in
$P_6$-free graphs~[SODA 2019, TALG 2022], and of Abrishami, Chudnovsky,
Pilipczuk, Rz\k{a}\.zewski, and Seymour on problems in $P_5$-free
graphs~[SODA~2021]. The key step is a new generalization of the framework of
potential maximal cliques. We show that instead of listing a large family of
potential maximal cliques, it is sufficient to only list their carvers: vertex
sets that contain the same vertices from the sought solution and have similar
separation properties.
