Effectiveness and strong graph indivisibility

TL;DR

This paper investigates the concept of strong graph indivisibility, characterizes which graphs possess this property, and analyzes the computational and logical strength of Cameron's theorem using tools from computability theory and reverse mathematics.

Contribution

It demonstrates that Cameron's theorem is effective up to computable presentations and explores its validity within the -model REC, highlighting the logical strength of the original proof.

Findings

Cameron's theorem is effective for computable presentations.

Partial results suggest the theorem holds in the -model REC.

The original proof relies on the stronger induction scheme I.

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 $\omega$-model $\mathsf{REC}$. We also establish that Cameron's original proof makes essential use of the stronger induction scheme $\mathsf{I}\Sigma^0_2$.