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

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 Hubička, Konečný, Todorčević and Zucker (announced at EUROCOMB 2025) that the random edge-labelled graph (the Fraïssé limit of the class of all finite complete edge-labelled graphs where the set of labels is countably infinite) does not have finite big Ramsey degrees. We then use the machinery of category theory to transport this result from the context of edge-labelled graphs to the context of groups. The main step in the process is the construction of a functor from the category of edge-labelled graphs and embeddings to the category of groups and group embeddings which takes finite graphs to finite groups. This makes is possible for us to build a subgroup of the Hall's universal group which encodes the random edge-labelled graph.