Large cliques or stable sets in graphs with no four‐edge path and no five‐edge path in the complement
Why this work is in the frame
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Bibliographic record
Abstract
Abstract Erdős and Hajnal [Discrete Math 25 (1989), 37–52] conjectured that, for any graph H , every graph on n vertices that does not have H as an induced subgraph contains a clique or a stable set of size n ɛ( H ) for some ɛ( H )>0. The Conjecture 1. known to be true for graphs H with | V ( H )|≤4. One of the two remaining open cases on five vertices is the case where H is a four‐edge path, the other case being a cycle of length five. In this article we prove that every graph on n vertices that does not contain a four‐edge path or the complement of a five‐edge path as an induced subgraph contains either a clique or a stable set of size at least n 1/6 . © 2011 Wiley Periodicals, Inc. J Graph Theory
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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 | 0.007 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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