We study cardinal invariants of systems of meager hereditary families of subsets of ω connected with the collapse of the continuum by Sacks forcing 𝕊 and we obtain a cardinal invariant 𝔥ω such that 𝕊 collapses the continuum to 𝔥ω and 𝔥≤𝔥ω≤𝔟. Applying the Baumgartner-Dordal theorem on preservation of eventually narrow sequences we obtain the consistency of 𝔥=𝔥ω<𝔟. We define two relations ≼*0 and ≼*1 on the set (ωω)Fin of finite-to-one functions which are Tukey equivalent to the eventual dominance relation of functions such that if ℱ⊆(ωω)Fin is ≼*1-unbounded, well-ordered by ≼*1, and not ≼*0-dominating, then there is a nonmeager p-ideal. The existence of such a system ℱ follows from Martin’s axiom. This is an analogue of the results of , [9,10] for increasing functions.
"Cardinal invariants and the collapse of the continuum by Sacks forcing." J. Symbolic Logic 73 (2) 711 - 727, June 2008. https://doi.org/10.2178/jsl/1208359068