Bibliographic record
Abstract
By the famous theorems of Rédei, a set of q points in AG(2, q) (respectively p points in AG(2, p), p prime) is either a line or it determines at least (respectively ) directions. We generalize these results on two fronts. First we provide bounds on the number of directions determined by a set of n≤q points in a general projective plane of order q. Secondly, given a dual n-arc in Π=PG(k, q) we consider Π as embedded in Σ=PG(k+1, q) where E=Σ–Π is the associated affine space. A collection of affine points is a transversal set of if any line incident with a k-fold point of is incident with at most one point of S. We reformulate Rédei’s results in the plane as results on transversal sets. In this setting we generalize Rédei’s theorems to higher dimensions. We also provide a new proof of a well known theorem on extending arcs in PG(k, q).
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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 | 0.003 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
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".