Hostname: page-component-cb9f654ff-p5m67 Total loading time: 0 Render date: 2025-09-09T19:04:42.828Z Has data issue: false hasContentIssue false
  • English
  • Français

The ultrafilter number and $\mathfrak {hm}$

Part of: Set theory

Published online by Cambridge University Press:  03 November 2021

Osvaldo Guzmán
Osvaldo Guzmán*
Affiliation:
Centro de Ciencias Matematicas, UNAM, Morelia, Michoacán, México
*
e-mail: oguzman@matmor.unam.mx
Get access Rights & Permissions [Opens in a new window]

Abstract

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 $\mathfrak {i<u},$ so the inequality $\mathfrak {hm<u}$ 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.

Keywords

Cardinal invariants of the continuumcontinous coloringsultrafiltersultrafilter numberiterated forcingMAD families

MSC classification

Primary: 03E17: Cardinal characteristics of the continuum
Secondary: 03E35: Consistency and independence results 03E05: Other combinatorial set theory

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 2 , April 2023 , pp. 494 - 530
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

The author was partially supported by a CONACyT grant A1-S-16164 and PAPIIT grant IN104220.

References

Abraham, U., Proper forcing . In Handbook of set theory . Vols. 1–3, Springer, Dordrecht, 2010, pp. 333–394.CrossRefGoogle Scholar
Bartoszyński, T. and Judah, H., Set theory: on the structure of the real line. A. K. Peters Ltd., Wellesley, MA, 1995.CrossRefGoogle Scholar
Blass, A., Combinatorial cardinal characteristics of the continuum. In Handbook of set theory . Vols. 1–3, Springer, Dordrecht, 2010, pp. 395–489.CrossRefGoogle Scholar
Brendle, J., Mad families and iteration theory. In Logic and algebra, Contemporary Mathematics, 302, American Mathematical Society, Providence, RI, 2002, pp. 1–31.Google Scholar
Chodounský, D., Fischer, V., and Grebík, J., Free sequences in $P(\omega )/ fin$ . Arch. Math. Logic 58(2019), nos. 7–8, 1035–1051.CrossRefGoogle Scholar
Chodounský, D. and Guzmán, O., There are no P-points in silver extensions. Israel J. Math. 232(2019), no. 2, 759–773.CrossRefGoogle Scholar
Geschke, S.. A dual open coloring axiom. Ann. Pure Appl. Logic 140(2006), no. 1, 40–51. Cardinal Arithmetic at work: the 8th Midrasha Mathematicae Workshop.CrossRefGoogle Scholar
Geschke, S., Goldstern, M., and Kojman, M., Continuous Ramsey theory on polish spaces and covering the plane by functions. J. Math. Logic 4(2004), no. 12, 109–145.CrossRefGoogle Scholar
Geschke, S., Kojman, M., Kubiś, W., and Schipperus, R., Convex decompositions in the plane and continuous pair colorings of the irrationals. Isr. J. Math. 131(2002), 285–317.CrossRefGoogle Scholar
Hrušák, M., Another $\ \diamond$ -like principle. 167(2001), no. 01, 277–289.Google Scholar
Ishiu, T., $\unicode{x3b1}$ -properness and axiom A. Fund. Math. 186(2005), no. 1, 25–37.CrossRefGoogle Scholar
Jorge, V. C.-C., Fischer, V., Guzmán, O., and Jaroslav, S.. Independent families and compact partitions. Preprint.Google Scholar
Shelah, S., CON( $u>i$ ). Arch. Math. Logic 31(1992), no. 6, 433–443.CrossRefGoogle Scholar
Shelah, S., Proper and improper forcing. In: Perspectives in mathematical logic, 2nd ed., Springer-Verlag, Berlin, 1998.Google Scholar
Shelah, S., Two cardinal invariants of the continuum $\ (d<a)$ and FS linearly ordered iterated forcing. Acta Math. 192(2004), no. 2, 187–223.CrossRefGoogle Scholar