3 problems

Consistency of the Boolean Prime Ideal Theorem with no Vitali set
Consistency conjecture. It is consistent with ZF plus the Axiom of Dependent Choices (DC) that BPI holds and there is no Vitali set in the real line.
openFrom papersaxiom-of-choice boolean-algebras set-theory
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3
Axiom of Choice equivalence for colourings of infinite connected graphs
In ZF, let a cardinal number be defined as an equivalence class under equinumerosity. For a graph, its chromatic index is the least cardinal number of colours in a proper edge-colo…
openFrom papersaxiom-of-choice graph-colourings set-theory
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6
Converse of the internal-automorphism criterion for Frucht's theorem
Let $M$ be a model of set theory satisfying the Axiom of Choice, and let $M'=M/\mathcal{F}$ be a permutation model inside $M$ . Suppose that Frucht's Theorem fails in $M'$ . An inter…
openFrom papersaxiom-of-choice graph-automorphisms permutation-models
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5

Consistency of the Boolean Prime Ideal Theorem with no Vitali set