# Coloring graphs on Euclidean spaces

## Jindrich Zapletal

*University of Florida / Czech Academy of Sciences*

June 7, 2022, 15:00 in S5

## Abstract

For n>0, let Gn be the graph connecting points in Rn of rational Euclidean distance. In ZFC, these graphs all have countable chromatic number. I proved that for every n>0, it is consistent with choiceless set theory that Gn has countable chromatic number while Gn+1 does not, showing that the coloring problem grows in difficulty with n. The proof depends on certain finite combinatorial features of Rn, and the talk will concentrate on these.