No Borel Connections for the Unsplitting Relations

Abstract

We prove that there is no Borel connection for non-trivial pairs of unsplitting relations. This was conjectured in [3].Mathematics Subject Classification: 03E15, 03E17, 03E35. Vojtáš [6] introduced a framework in which cardinal characteristics of the continuum can be regarded as norms of corresponding relationsA Galois-Tukey connection from a relation B to a relation A, which we call as in [1] a morphism from A to B, is a pair of functions (α, β) such thatIf there is a morphism from A to B, then B ≤ A , and indeed the proofs of the inequalities usually exhibit morphisms between the corresponding relations. We deal with the unsplitting relations: For n ≥ 1, we haver 2 is the usual unsplitting number. It is easy to see that r m = r n for m, n ≥ 2.In [3] it was proved that there is no morphism for the sharp unsplitting relations with Baire measurable first component, and it was conjectured that the same is true for the ordinary unsplitting relations. In this paper we prove this conjecture, under the additional premise that α and β are Lebesgue measurable.The present work is built partly upon techniques from the mentioned work on the sharp unsplitting relations, to which we add some additional steps mainly coming from 1) The author was supported by a Minerva fellowship. 2)

Cardinal characteristics and countable Borel equivalence relations

“…Borel morphisms have been studied in [Bla96], [Mil02] and [CSM13]. The latter article provides a diagram of all the characteristics in Figure 3 with respect to Borel morphisms.…”

Pair-splitting, pair-reaping and cardinal invariants ofFσ-ideals

“…However it might be difficult. In [11] and [12], Mildenberger introduced another variation of reaping numbers r n and r n = r m (= r) holds for n, m ∈ ω but it is proved that there are no nice Galois-Tukey connections between Mildenberger's reaping numbers. .…”

Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

“…Since Blass's initial study, however, there have been just a couple of results on Borel morphisms. Blass's conjecture concerning r n was established by Mildenberger, who showed in [Mil02] that there are no such Borel morphisms. Another step was taken in [PR95], where the authors show that after suitably coding the null and meager ideals, all of the inequalities in Cicho ń's diagram are witnessed by Borel (in fact continuous) morphisms.…”