Colourful theorems and indices of homomorphism complexes
Why this work is in the frame
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Bibliographic record
Abstract
We extend the colourful complete bipartite subgraph theorems of [G. Simonyi, G. Tardos, Local chromatic number, Ky Fan's theorem, and circular colorings, Combinatorica 26 (2006), 587--626] and [G. Simonyi, G. Tardos, Colorful subgraphs of Kneser-like graphs, European J. Combin. 28 (2007), 2188--2200] to more general topological settings. We give examples showing that the hypotheses are indeed more general. We use our results to show that the topological bounds on chromatic numbers of digraphs with tree duality are at most one better than the clique number. We investigate combinatorial and complexity-theoretic aspects of relevant order-theoretic maps.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.000 | 0.006 |
| Scholarly communication | 0.000 | 0.005 |
| Open science | 0.009 | 0.004 |
| 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