The ultrafilter number and $\mathfrak {hm}$
Canadian journal of mathematics, Tome 75 (2023) no. 2, pp. 494-530
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The cardinal invariant $\mathfrak {hm}$ is defined as the minimum size of a family of $\mathsf {c}_{\mathsf {min}}$-monochromatic sets that cover $2^{\omega }$ (where $\mathsf {c}_{\mathsf {min}}( x,y) $ is the parity of the biggest initial segment both x and y have in common). We prove that $\mathfrak {hm}=\omega _{1}$ holds in Shelah’s model of $\mathfrak {i so the inequality $\mathfrak {hm is consistent with the axioms of $\mathsf {ZFC}$. This answers a question of Thilo Weinert. We prove that the diamond principle $\mathfrak {\Diamond }_{\mathfrak {d}}$ also holds in that model.
Mots-clés :
Cardinal invariants of the continuum, continous colorings, ultrafilters, ultrafilter number, iterated forcing, MAD families
Guzmán, Osvaldo. The ultrafilter number and $\mathfrak {hm}$. Canadian journal of mathematics, Tome 75 (2023) no. 2, pp. 494-530. doi: 10.4153/S0008414X21000614
@article{10_4153_S0008414X21000614,
author = {Guzm\'an, Osvaldo},
title = {The ultrafilter number and $\mathfrak {hm}$},
journal = {Canadian journal of mathematics},
pages = {494--530},
year = {2023},
volume = {75},
number = {2},
doi = {10.4153/S0008414X21000614},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X21000614/}
}
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