Year: 2024
Type: preprint
Abstract: A relational structure is \emph{strongly indivisible} if for every partition $M = X_0 \sqcup X_1$, the induced substructure on $X_0$ or $X_1$ is isomorphic to $\mathcal{M}$. Cameron (1997) showed that a graph is strongly indivisible if and only if it is the complete graph, the completely disconnected graph, or the random graph. We analyze the strength of Cameron's theorem using tools from computability theory and reverse mathematics. We show that Cameron's theorem is is effective up to computable presentation, and give a partial result towards showing that the full theorem holds in the $ω$-model $\mathsf{REC}$. We also establish that Cameron's original proof makes essential use of the stronger induction scheme $\mathsf{I}Σ^0_2$.
Source: arXiv (Cornell University)
Authors: Dzhafarov, Damir D., Solomon, Reed, Volpi, Andrea
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4404988290