October 15
Yuxin Zhou, University of Florida
Color isosceles triangles countably in $\mathbb{R}^2$ and but not in $\mathbb{R}^3$

Let $n>1$ be a natural number, let $\Gamma_n$ be the hypergraph of isosceles triangles in $\mathbb{R}^n$. Under the axiom of choice, the existence of a countable coloring for $\Gamma_n$ is true for every $n$. Without the axiom of choice, the coloring problems will be hard to answer. We often expect the case that the countable chromatic number of one hypergraph doesn't imply the one for another. With an inaccessible cardinal, 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.