Graph-Coloring and Choice a Note on a Note by Shelah and Soifer

Abstract

Sharpening results of Shelah and Soifer we will show that in ZF set theory the Shelah-Soifer Graph G has the following properties: 1. If AC(2), the axiom of choice for families of 2-element sets, or AC(R), the axiom of choice for families of non-empty subsets of R, hold, then G is 2-colorable. 2. For every cardinal c and for every c-coloring f : G → c each of the coloring sets f −1(i) is either non-measurable or of measure zero. 3. For every 2-coloring f : G → 2 both coloring sets f −1(0) and f −1(1) are non-measurable. 4. For every n-coloring f : G → n where n < ℵ0, there exists a non-measurable coloring set f −1(i). 5. If CC(R), the axiom of choice for countable families of non-empty subsets of R, holds, then for every ℵ0-coloring f : G → ℵ0 there exists a non-measurable coloring set f −1(i). 6. If AD, the axiom of determinateness, holds then, for any cardinal c ≤ ℵ0, the graph G has no c-coloring.