On-line Ramsey numbers for paths and stars
Why this work is in the frame
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Bibliographic record
Abstract
Graphs and Algorithms We study on-line version of size-Ramsey numbers of graphs defined via a game played between Builder and Painter: in one round Builder joins two vertices by an edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number r(H) of the graph H. We determine exact values of r(H) for a few short paths and obtain a general upper bound r(Pn) ≤ 4n −7. We also study asymmetric version of this parameter when one of the target graphs is a star Sn with n edges. We prove that r(Sn, H) ≤ n*e(H) when H is any tree, cycle or clique
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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