The Rado graph is outside the displayed hypothesis class

Exact statement

The pure Rado graph has no distinguished invariant linear-order relation, and the ordered random graph fails the required labelled-WQO condition: its age contains the chordless cycles $C_n$ as an induced-subgraph antichain. Thus neither structure satisfies the displayed conjunction of hypotheses.

StatusProved by an induced-cycle antichain; consistent with the later undecidability theorem

External reviewNo documented external or specialist review of this FPRD exposition is recorded.

Context

Homogeneity alone is not enough for the finite-basis or Gröbner conclusions.

Proof or evidence

No chordless cycle is an induced subgraph of a different chordless cycle. The 2025 preprint independently proves undecidable equivariant ideal membership for the Rado-graph action by its infinite-path criterion.

Rado-graph hostile boundary →

Verification notes

Both the missing-order boundary and the stronger ordered-random-graph antichain were checked; no undecidability claim was transferred to the ordered expansion.

Limitations

This excludes a proposed example; it does not resolve the weaker labelled-WQO implication for structures that actually satisfy the hypotheses.

Open work

Keep dense rational order as the positive example and do not list the Rado graph as satisfying both hypotheses.