Bibliographic record
Abstract
A Gallai coloring of a complete graph is a coloring of the edges without rainbow triangles (all edges colored differently). Given a graph H and a positive integer k, the Gallai Ramsey number GRk(H) is the minimum integer N such that every Gallai k-coloring of KN contains a monochromatic copy of H. A uniform coloring of a complete multipartite graph is a coloring of the edges such that the edges between any two parts receive the same color. Given a graph H and positive integers k and ℓ, the ℓ-uniform Ramsey number Rkℓ(H) is the minimum integer N such that any uniform k-coloring of any complete multipartite graph on N vertices with each part of cardinality no more than ℓ, contains a monochromatic copy of H. Let Km,n denote a complete bipartite graph. Wu et al. conjectured that GRk(Km,n)=(n−1)(k−2)+R21(Km,n) if R21(Km,n)≥3m−2, where k≥3 and m≥n≥2. In this paper, we show that GRk(Km,n)=(n−1)(k−2)+R2m−1(Km,n) for all integers k≥3 and m≥n≥2. Based on this result, we improve the known general bounds for GRk(Km,n), and confirm the conjecture for some complete bipartite graphs.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.003 | 0.002 |
| Scholarly communication | 0.009 | 0.004 |
| Open science | 0.002 | 0.006 |
| Research integrity | 0.003 | 0.002 |
| Insufficient payload (model declined to judge) | 0.425 | 0.414 |
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; the direct Gemma label and the distilled Codex classifier agree on what is shown here.
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".