On the Existence of a Point Subset with 4 or 5 Interior Points
Bibliographic record
Abstract
An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k 1, let h(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least h(k) interior points has a subset of points containing k or k +1 interior points. We proved that h(3) = 3 in an earlier paper. In this paper we prove that h(4) = 7. 2 Introduction Throughout the paper we consider only planar point sets in which no three points are collinear. For such a point set P we distinguish its vertices which lie on the boundary of its convex hull, from the remaining interior points. In 1935, Erdos and Szekeres [2] proved that for every t 3 there is a number e-mail address: avis@mutt.cs.mcgill.ca y e-mail address: hosono@scc.u-tokai.ac.jp z e-mail address: qzg00130@scc.u-tokai.ac.jp 1 f(t) such that every set P of at least f(t) points, no three collinear, contains a subset of points whose conv...
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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 | 0.019 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.003 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 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".