Off-Diagonal Commonality of Graphs via Entropy
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Bibliographic record
Abstract
.A graph \(H\) is common if the limit as \(n\to \infty\) of the minimum density of monochromatic labeled copies of \(H\) in an edge coloring of \(K_n\) with red and blue is attained by a sequence of quasirandom colorings. We apply an information-theoretic approach to show that certain graphs obtained from odd cycles and paths via gluing operations are common. In fact, for every pair \((H_1,H_2)\) of such graphs, there exists \(p\in (0,1)\) such that an appropriate linear combination of red copies of \(H_1\) and blue copies of \(H_2\) is minimized by a quasirandom coloring in which \(p\binom{n}{2}\) edges are red; such a pair \((H_1,H_2)\) is said to be \((p,1-p)\) -common. Our approach exploits a strengthening of the common graph property for odd cycles that was recently proved using Schur convexity. We also exhibit a \((p,1-p)\) -common pair \((H_1,H_2)\) such that \(H_2\) is uncommon.KeywordscommongraphhomomorphismSidorenkooff-diagonalRamseyMSC codes05C5505C35
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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 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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