Bibliographic record
Abstract
Abstract An ‐star is a complete bipartite graph . An ‐star system of order , is a partition of the edges of the complete graph into ‐stars. An ‐star system is said to be ‐colourable if its vertex set can be partitioned into sets (called colour classes) such that no ‐star is monochromatic. The system is ‐chromatic if is ‐colourable but is not ‐colourable. If every ‐colouring of an ‐star system can be obtained from some ‐colouring by a permutation of the colours, we say that the system is uniquely ‐colourable. In this paper, we first show that for any integer , there exists a ‐chromatic 3‐star system of order for all sufficiently large admissible . Next, we generalize this result for ‐star systems for any . We show that for all and , there exists a ‐chromatic ‐star system of order for all sufficiently large such that (mod ). Finally, we prove that for all and , there exists a uniquely ‐chromatic ‐star system of order for all sufficiently large such that (mod ).
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 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".