Delaunay graphs of point sets in the plane with respect to axis‐parallel rectangles
Bibliographic record
Abstract
Abstract Given a point set P in the plane, the Delaunay graph with respect to axis‐parallel rectangles is a graph defined on the vertex set P , whose two points p , q ∈ P are connected by an edge if and only if there is a rectangle parallel to the coordinate axes that contains p and q , but no other elements of P . The following question of Even et al. (SIAM J Comput 33 (2003) 94–136) was motivated by a frequency assignment problem in cellular telephone networks: Does there exist a constant c > 0 such that the Delaunay graph of any set of n points in general position in the plane contains an independent set of size at least cn ? We answer this question in the negative, by proving that the largest independent set in a randomly and uniformly selected point set in the unit square is O ( n log 2 log n /log n ), with probability tending to 1. We also show that our bound is not far from optimal, as the Delaunay graph of a uniform random set of n points almost surely has an independent set of size at least c n log log n /(log n log log log n ). We give two further applications of our methods: (1) We construct two‐dimensional n ‐element partially ordered sets such that the size of the largest independent sets of vertices in their Hasse diagrams is o ( n ). This answers a question of Matoušek and Přívětivý (Combinat Probab Comput 15 (2006) 473–475) and improves a result of Kříž and Nešetřil (Order 8 (1991) 41–48). (2) For any positive integers c and d , we prove the existence of a planar point set with the property that no matter how we color its elements by c colors, we find an axis‐parallel rectangle containing at least d points, all of which have the same color. This solves an old problem from the work of Brass et al. (Research Problem in Discrete Geometry Springer‐Verlag, New York, 2005). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".