Restricted size Ramsey number for 2K <sub>2</sub> versus disconnected graphs of order six
Bibliographic record
Abstract
Abstract Given simple graphs F , G , and H . We say F arrows ( G , H ) if for any red-blue coloring of the edge of F , we find either a red-colored graph G or a blue-colored graph H . The Ramsey number r ( G , H ) is the smallest positive integer r such that a complete graph K r arrows ( G , H ). The size Ramsey number is the smallest positive integer <?CDATA $\hat{r}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> such that a graph F with the size of <?CDATA $\hat{r}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> arrows ( G , H ). The restricted size Ramsey number is the smallest positive integer r ∗ such that a graph F , of order r ( G , H ) with the size of r ∗ , arrows ( G , H ). In this paper we give the restricted size Ramsey number of a matching of two edges and any disconnected graphs of order six with no isolates.
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How this classification was reachedexpand
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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".