For \(n\) a positive natural number, let \(\Gamma_n\) be the hypergraph of isosceles triangles on \(\mathbb{R}^n\). Under the axiom of choice, the existence of a countable coloring for \(\Gamma_n\) holds for every \(n\). Without the axiom of choice, the chromatic numbers may or may not be countable. With an inaccessible cardinal assumption, there is a model of ZF+DC in which \(\Gamma_2\) has countable chromatic number while \(\Gamma_3\) has uncountable chromatic number. This result is obtained by a balanced forcing over the symmetric Solovay model.
Distinguish chromatic numbers for isosceles triangles in choiceless set theory
14.12.2021 17:00 - 17:50
Organiser:
KGRC
Location:
Zoom Meeting