The saturation of club guessing ideals

We prove that it is consistent that there exists a saturated tail club guessing ideal on ω 1 which is not a restriction of the non-stationary ideal. Two proofs are presented. The first one uses a new forcing axiom whose consistency can be proved from a supercompact cardinal. The resulting model can satisfy either CH or 2 ℵ 0 = ℵ 2 . The second one is a direct proof from a Woodin cardinal, which gives a witnessing model with CH.