Call a colouring of graph distinguishing if the only automorphism which preserves it is identity. We investigate role Axiom Choice in existence certain proper or colourings both vertex and edge variants with special emphasis on locally finite connected graphs. show that every has at most countable number colours Kőnig's Lemma holds. This statement holds for colourings. Furthermore, we improvabl...

For certain weak versions of the Axiom of Choice (most notably, the Boolean Prime Ideal theorem), we obtain equivalent formulations in terms of partial orders, and filter-like objects within them intersecting certain dense sets or antichains. This allows us to prove some consequences of the Boolean Prime Ideal theorem using arguments in the style of those that use Zorn’s Lemma, or Martin’s Axiom.

We exhibit an interpretation of the Extended Set Theory proposed by Dave Childs in classical Zermelo-Fraenkel set theory with the axiom of choice and an axiom asserting the existence of arbitrarily large inaccessible cardinals. In particular, if the existence of arbitrarily large inaccessible cardinals is consistent with ZFC, then Childs’s Extended Set Theory is also consistent.

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In previous papers (J. Combin Theory Ser. A 103 (2003) 387) and (J. Combin. Theory Ser. A 105 (2004) 359) Saharon Shelah and I formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory.We also constructed examples of a distance graph on the real line R and differe...