Nathanson Heights and the CSS Conjecture for Cayley Graphs
Bibliographic record
Abstract
Let G be a finite directed graph, \beta(G) the minimum size of a subset X of edges such that the graph G'=(V, E / X) is directed acyclic and \gamma(G) the number of pairs of nonadjacent vertices in the undirected graphobtained from G by replacing each directed edge with an undirected edge. Chudnovsky, Seymour and Sullivanproved that if G is triangle-free, then \beta(G)\leq\gamma(G). They conjectured a sharper bound (so called the “CSS conjecture”) that \beta(G)\leq\gamma(G)/2. Nathanson and Sullivan verified this conjecture for the directed Cayleygraph Cay(Z/NZ, EA) whose vertex set is the additive group Z/NZ and whose edge set EA is determined byEA = {(x, x +a) : x \in Z/NZ, a \in A} when N is prime and |A| \leq (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, EA) to any positiveinteger N and |A|\leq (N-1)/4.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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 source (direct Gemma or distilled Codex), 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".