A counterexample to a conjecture about triangle-free induced subgraphs of graphs with large chromatic number
Bibliographic record
Abstract
We prove that for every $n$, there is a graph $G$ with $\chi(G) \geq n$ and $\omega(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $\omega(H) \leq 2$ satisfies $\chi(H) \leq 4$. This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.
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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 | 0.002 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.002 | 0.007 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.008 | 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".