Bibliographic record
Abstract
Let G be a finite directed graph, (G) the minimum size of a subset X of edges such that the graph G = (V, E X) is directed acyclic and (G) the number of pairs of nonadjacent vertices in the undirected graph obtained from G by replacing each directed edge with an undirected edge. Chudnovsky, Seymour and Sullivan proved that if G is triangle-free, then (G) (G). They conjectured a sharper bound (so called the "CSS conjecture") that (G) (G)/2. Nathanson and Sullivan verified this conjecture for the directed Cayley graph Cay(Z/NZ, E A ) whose vertex set is the additive group Z/NZ and whose edge set E A is determined by E A = {(x, x + a) : x Z/NZ, a A} when N is prime and |A| (N -1)/4 by introducing "height". In this work, we extend the definition of height and apply to answer the CSS conjecture for Cay(Z/NZ, E A ) to any positive integer N and |A| (N -1)/4.
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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 | 0.044 | 0.039 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.003 | 0.000 |
| Research integrity | 0.000 | 0.005 |
| 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 itClassification
machine, unvalidatedMachine predicted; both teacher heads 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".