Lower Bounds for the Cop Number when the Robber is Fast
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Bibliographic record
Abstract
We consider a variant of the Cops and Robbers game where the robber can move t edges at a time, and show that in this variant, the cop number of a d -regular graph with girth larger than 2 t +2 is Ω( d t ). By the known upper bounds on the order of cages, this implies that the cop number of a connected n -vertex graph can be as large as Ω( n 2/3 ) if t ≥ 2, and Ω( n 4/5 ) if t ≥ 4. This improves the Ω( $n^{\frac{t-3}{t-2}}$ ) lower bound of Frieze, Krivelevich and Loh (Variations on cops and robbers, J. Graph Theory , to appear) when 2 ≤ t ≤ 6. We also conjecture a general upper bound O ( n t / t +1 ) for the cop number in this variant, generalizing Meyniel's conjecture.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it