Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path
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Bibliographic record
Abstract
Abstract A graph G is H -free if it has no induced subgraph isomorphic to H . We prove that a $$P_5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>P</mml:mi><mml:mn>5</mml:mn></mml:msub></mml:math> -free graph with clique number $$\omega \ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ω</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math> has chromatic number at most $$\omega ^{\log _2(\omega )}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>ω</mml:mi><mml:mrow><mml:msub><mml:mo>log</mml:mo><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>ω</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup></mml:math> . The best previous result was an exponential upper bound $$(5/27)3^{\omega }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>5</mml:mn><mml:mo>/</mml:mo><mml:mn>27</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mn>3</mml:mn><mml:mi>ω</mml:mi></mml:msup></mml:mrow></mml:math> , due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for $$P_5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>P</mml:mi><mml:mn>5</mml:mn></mml:msub></mml:math> , which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for $$P_5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>P</mml:mi><mml:mn>5</mml:mn></mml:msub></mml:math> -free graphs, and our result is an attempt to approach that.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 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