The diamond principle conjecture for mad families of block subspaces

From papers

Let FF be a countable field, and let (ωω,=)\diamondsuit(\omega^\omega,=^\infty) denote the parameterized diamond principle appearing in the source. Diamond principle conjecture. If (ωω,=)\diamondsuit(\omega^\omega,=^\infty) holds, then there is a mad family of block subspaces of cardinality 1\aleph_1, and consequently avec,F=1\mathfrak{a}_{\mathrm{vec},F}=\aleph_1. This is presented as the suspected correct diamond principle for determining when avec,F=1\mathfrak{a}_{\mathrm{vec},F}=\aleph_1, in contrast with the corresponding principle for ordinary mad families; its status is not resolved in the supplied text.

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Sources & referencesView supporting material

Primary source

Iian B. Smythe, “Madness in vector spaces”, arXiv:1712.00057 (2019).

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