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We study cardinal invariants of systems of meager hereditary families of subsets of ω connected with the collapse of the continuum by Sacks forcing 
and we obtain a cardinal invariant 
such that collapses the continuum to and 
. Applying the Baumgartner-Dordal theorem on preservation of eventually narrow sequences we obtain the consistency of 
. We define two relations 
and 
on the set (ωω)Fin of finite-to-one functions which are Tukey equivalent to the eventual dominance relation of functions such that if 
-unbounded, well-ordered by , and not -dominating, then there is a nonmeager p-ideal. The existence of such a system 
follows from Martin's axiom. This is an analogue of the results of [3], [9, 10] for increasing functions.
