A note on the relation between Hartnell’s firefighter problem and growth of groups
Bibliographic record
Abstract
The firefighter game problem on locally finite connected graphs was introduced by Bert Hartnell [6]. The game on a graph G can be described as follows: let f n be a sequence of positive integers; an initial fire starts at a finite set of vertices; at each (integer) time n ≥ 1 , f n vertices which are not on fire become protected, and then the fire spreads to all unprotected neighbors of vertices on fire; once a vertex is protected or is on fire, it remains so for all time intervals. The graph G has the f n -containment property if every initial fire admits an strategy that protects f n vertices at time n so that the set of vertices on fire is eventually constant. If the graph G has the containment property for a sequence of the form f n = C n d , then the graph is said to have polynomial containment . In [5], it is shown that any locally finite graph with polynomial growth has polynomial containment; and it is remarked that the converse does not hold. That article also raised the question of whether the equivalence of polynomial growth and polynomial containment holds for Cayley graphs of finitely generated groups. In this short note, we remark how the equivalence holds for elementary amenable groups and for non-amenable groups from results in the literature.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.026 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.004 | 0.008 |
| Scholarly communication | 0.004 | 0.018 |
| Open science | 0.003 | 0.007 |
| Research integrity | 0.005 | 0.010 |
| Insufficient payload (model declined to judge) | 0.013 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".