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

2026/07/26 by Dragan Mašulović, Veljko Toljić
#math.CO

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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 Fra"ıssé 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.

Citations

• On Big Ramsey degrees of universal ω-edge-labeled hypergraphs
• Fraïssé's Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition
• Finite big Ramsey degrees in universal structures

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• On Big Ramsey degrees of universal ω-edge-labeled hypergraphs
• Finite big Ramsey degrees in universal structures
• Big Ramsey degrees using parameter spaces
• Counting big Ramsey degrees of the homogeneous and universal K₄-free graph
• Big Ramsey degrees in universal inverse limit structures