Madness in Vector Spaces

Abstract

We consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and that the "spectrum" of cardinalities of mad families of subspaces can be made arbitrarily large, in analogy to results for mad families on ω. We apply the author's local Ramsey theory for vector spaces [32] to give partial results concerning their definability. The author would like to thank the anonymous referee for many helpful comments and corrections. 1 2 IIAN B. SMYTHE One way of addressing question 1 is to determine the value of the cardinal invariant a = min{|A| : A is an infinite mad family}.This could mean which ℵ α is such that a = ℵ α , or how a relates to other well-studied cardinal invariants (see [5] or [36]) between ℵ 1 and c. By our comments above, ℵ 1 ≤ a ≤ c, and a modification of that diagonalization argument shows that b ≤ a, where b is the minimum size of an unbounded family of functions ω → ω (see [ibid.]). However, the value of a cannot be decided in ZFC: both the Continuum Hypothesis CH and Martin's Axiom MA (see [21] or [22]) imply that a = c, and thus, consistently ℵ 1 < a = c, while Kunen [21] showed that in the model obtained by adding ℵ 2 -many Cohen reals to a model of CH, ℵ 1 = a < c = ℵ 2 . In [19], Hrušák showed 1 that the latter also holds in the models obtained by adding ℵ 2 -many Sacks reals iteratively or "side-by-side" to a model of CH.A more sophisticated version of question 1 might ask for the "spectrum" of cardinalities between ℵ 1 and c that mad families can posses. This was first addressed by Hechler [14], who produced a method for obtaining arbitrarily large continuum and, simultaneously, mad families of all cardinalities κ for ℵ 1 ≤ κ ≤ c. While beyond the scope of our investigations here, these questions have been the focus of much deep work in recent decades, notably Brendle's [6], which establishes the consistency of a = ℵ ω , Shelah's [29], which establishes the consistency of d < a, and Shelah and Spinas' [30], which gives a nearly-sharp characterization of possible mad spectra.Question 2 above seeks to understand to what extent the nonconstructive methods used to obtain mad families are necessary. A result of Mathias [23] says that an infinite mad family can never be analytic (i.e., a continuous image of a Borel set). Under large cardinal hypotheses, this can be pushed further to show that there are no definable mad families at all, in the sense that there are none in L(R) (see [9], [23], [35], and for a consistency result without large cardinals, [16]). Mathias' result is also sharp; Miller [24] proved that there is a coanalytic (i.e., the complement of an analytic set) mad family assuming V = L, work later refined by Törnquist [34].This article is concerned with an analogue of mad families arising in vector spaces. Throughout, E will be a countably infinite-dimensional vector space over a countable (possibly finite) field F .D...

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

“…Consequently, this cardinal invariant is ℵ1 in the Miller model. This verifies a conjecture of the author from [14].…”

“…The basic properties of mad families of block subspaces and a vec,F can be found in [14]. In particular, a vec,F is uncountable (Proposition 2.5 in [14]).…”

“…When |F | = 2, our setting corresponds exactly to the study of combinatorial subspaces in FIN, the set of finite nonempty subsets of N. The resulting cardinal invariant, a FIN , was investigated by Brendle and García Ávila [5], who proved the following lower bound, which the author observed could be extended to a vec,F for general F : Theorem 1.1 (Corollary 2.11 in [14]). non(M) ≤ a vec,F .…”

“…It is shown in [14] that a vec,F is small in the Cohen and (iterated or side-by-side) Sacks models. It was conjectured in [14] that this is also the case in the Miller model (described at the end of §2 below), and moreover, that a certain parametrized ♦ principle, in the sense of Moore, Hrušák, and Džamonja [12], suffices. It is this conjecture that we verify in Theorem 3.1 below.…”

Parametrizing the Ramsey theory of block sequences I: Discrete vector spaces

“…Let A be a nonempty perfect almost disjoint collection of infinite-dimensional subspaces of E, i.e., for any distinct X, Y ∈ A, X ∩ Y is finite-dimensional. Such a family exists, see Proposition 2.2 of [35]. For each X ∈ T , let T X : E → X be an isomorphism.…”