What was already known
Frankl, Peng, Rödl and Talbot asked in 2007 whether such intervals exist. No positive-length interval for r≥3 was known; even positive Hausdorff dimension was open.
AI involvement
The authors credit ChatGPT 5.4 Pro and Thinking with suggesting the internal Ramanujan-graph gadget. Liu and Pikhurko developed and explained the surrounding constructions and spectral proof.
The source names a specific model-suggested mathematical idea central to the final construction while expressly crediting the other main ideas to the authors.
Importance
The model-suggested gadget supplies a central ingredient in the interval construction.
Limitations
The outer, recursive and lifting constructions are credited to Liu and Pikhurko, whose project predates the model suggestion. The paper gives proofs, but a completed check of this specific argument is not documented.
Technical verification
The paper includes detailed proofs of Construction 3.2, Lemma 3.3 and the interval theorem, but no documented completed human or machine check of the assessed gadget-to-theorem argument. Reported is the evidence default, not a finding that the proof failed.
Human understanding
The human authors explain how the Ramanujan spectral bound controls the gadget Lagrangian and how the resulting slowly convergent sequence feeds the recursive interval construction.
Correction and publication history
The model-suggested gadget supplies a central ingredient in the interval construction.
Read this published revision