GENERIC TREES

We continue the investigation of the Laver ideal l(0) and Miller ideal m(0) started in [GJSp] and [GRShSp]; these are the ideals on the Bair e space associated with Laver forcing and Miller Forcing. We solve sev eral open problems from these papers. The main result is the construct ion of models for I < add(l(0)), p < add(m(0)), where add denotes tile additivity coefficient of air ideal. For this we construct amoeba for cings for these forcings which do not add Cohen reals. We show that c = omega(2) implies add(m(0)) less than or equal to h. We show that b = c, d = c implies cov(l(0)) less than or equal to h(+), cov(m(0)) less than or equal to h(+) respectively. Here cov denotes the covering coe fficient. We also show that in the Cohen model cov(m(0)) < d holds. Fi nally we prove that Cohen forcing does not add a superperfect tree of Cohen reals.