Different covering numbers of compact tree ideals

We investigate the covering numbers of some ideals on \({^{\omega }}{2}{}\) $_{}^{\omega }2$ associated with tree forcings. We prove that the covering of the Sacks ideal remains small in the Silver and uniform Sacks model, respectively, and that the coverings of the uniform Sacks ideal and the Mycielski ideal, \({\mathfrak {C}_{2}}\) ${\mathfrak{C}}_2$ , remain small in the Sacks model.