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Mad families of vector subspaces and the smallest nonmeager set of reals

Volume 163 / 2021

Iian B. Smythe Colloquium Mathematicum 163 (2021), 15-22 MSC: Primary 03E17; Secondary 15A03. DOI: 10.4064/cm8044-12-2019 Published online: 20 May 2020


We show that a parametrized $\diamondsuit $ principle, corresponding to the uniformity of the meager ideal, implies that the minimum cardinality of an infinite maximal almost disjoint family of block subspaces in a countable vector space is $\aleph _1$. Consequently, this cardinal invariant is $\aleph _1$ in the Miller model. This verifies a conjecture of the author from his previous article Madness in vector spaces (J. Symbolic Logic, 2019).


  • Iian B. SmytheDepartment of Mathematics
    Rutgers University – New Brunswick
    110 Frelinghuysen Road
    Piscataway, NJ 08854, U.S.A.

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