Hall's universal group does not have finite big Ramsey degrees

Dragan Mašulović; Veljko Toljić

Hall's universal group does not have finite big Ramsey degrees

Combinatorics 2026-07-26 v1

Authors: Dragan Mašulović , Veljko Toljić

Abstract

In this paper we show that the Hall's universal group does not have finite big Ramsey degrees. Our strategy consists of piggybacking on the recent result of Hubi\v{c}ka, Kone\v{c}n\'y, Todor\v{c}evi\'c and Zucker (announced at EUROCOMB~2025) that the Fra\"{\i}ss\'e limit of the class of all finite complete edge-labelled graphs with where the set of labels is countably infinite does not have finite big Ramsey degrees. We then use our categorical machinery to transport their result from the context of edge-lebelled graphs to the context of groups.

Keywords

Hall's universal group big Ramsey degrees Fraisse limit edge-labelled graphs categorical machinery

Cite

@article{arxiv.2607.23618,
  title  = {Hall's universal group does not have finite big Ramsey degrees},
  author = {Dragan Mašulović and Veljko Toljić},
  journal= {arXiv preprint arXiv:2607.23618},
  year   = {2026}
}