Abstract
We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal \({{\mathcal {S}}}{{\mathcal {N}}}\). As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that \(\mathrm {non}({{\mathcal {S}}}{{\mathcal {N}}})<\mathrm {cov}({{\mathcal {S}}}{{\mathcal {N}}})<\mathrm {cof}({{\mathcal {S}}}{{\mathcal {N}}})\), which is the first consistency result where more than two cardinal invariants associated with \({{\mathcal {S}}}{{\mathcal {N}}}\) are pairwise different. Another consequence is that \({{\mathcal {S}}}{{\mathcal {N}}}\subseteq s^0\) in ZFC where \(s^0\) denotes Marczewski’s ideal.
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Notes
Obvious lower bounds of \(\mathrm{cof}({{\mathcal {S}}}{{\mathcal {N}}})\) are \(\mathrm{cov}({\mathcal {N}})\) and \(\mathrm{cov}({\mathcal {M}})\) because of (SN2) and (SN3), respectively.
Also \(\mathrm{non}({{\mathcal {S}}}{{\mathcal {N}}})\le \mathrm{non}(s^0)\), but \(\mathrm{non}(s^0)={\mathfrak {c}}\) because \([2^\omega ]^{<{\mathfrak {c}}}\subseteq s^0\).
In Yorioka’s original result it is further assumed that \({\mathfrak {d}}=\mathrm{cov}({\mathcal {M}})=\kappa \), but this redundant because \(\mathrm {minadd}\le \mathrm{add}({\mathcal {M}})\le \mathrm{cov}({\mathcal {M}})\le {\mathfrak {d}}\le \mathrm{cof}({\mathcal {M}})\le \mathrm {supcof}\), as pointed out in the paragraph preceding the theorem.
A poset \({\mathbb {P}}\) is strongly \(\omega ^\omega \)-bounding if for any \(p\in {\mathbb {P}}\) and any \({\mathbb {P}}\)-name \({\dot{x}}\) of a function from \(\omega \) into the ground model, there are a function f from \(\omega \) into the finite sets and some \(q\le p\) that forces \({\dot{x}}(n)\in f(n)\) for any \(n<\omega \).
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This work was supported by the Austrian Science Fund (FWF) P30666 and the DOC Fellowship of the Austrian Academy of Sciences at the Institute of Discrete Mathematics and Geometry, TU Wien (first author), the Grant-in-Aid for Early Career Scientists 18K13448, Japan Society for the Promotion of Science (second author), and by the Grant No. IN202010, Dirección de Tecnología e Investigación, Institución Universitaria Pascual Bravo (second and third authors)
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Cardona, M.A., Mejía, D.A. & Rivera-Madrid, I.E. The covering number of the strong measure zero ideal can be above almost everything else. Arch. Math. Logic 61, 599–610 (2022). https://doi.org/10.1007/s00153-021-00808-0
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DOI: https://doi.org/10.1007/s00153-021-00808-0