Large cliques or stable sets in graphs with no four‐edge path and no five‐edge path in the complement
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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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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 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".