Effectiveness and strong graph indivisibility

A relational structure is \emph{strongly indivisible} if for every partition $M=X_0\bigsqcup X_1$, 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 $\mathsf{R}\mathsf{E}\mathsf{C}$. We also establish that Cameron’s original proof makes essential use of the stronger induction scheme $\mathsf{I}{\mathrm{\Sigma }}_2^0$.