Design theory and some forbidden configurations
Bibliographic record
Abstract
Abstract In this paper we relate ‐designs to a forbidden configuration problem in extremal set theory. Let denote a column of 1's on top of 0's. Let denote the matrix consisting of rows of 1's and rows of 0's. We consider extremal problems for matrices avoiding certain submatrices. Let be a (0, 1)‐matrix forbidding any submatrix . Assume is ‐rowed and only columns of sum are allowed to be repeated. Assume that has the maximum number of columns subject to the given restrictions. Assume is sufficiently large. Then has each column of sum and exactly once and, given the appropriate divisibility condition, the columns of sum correspond to a ‐design with block size and parameter . The proof derives a basic upper bound on the number of columns of by a pigeonhole argument and then a careful argument, for large m, reduces the bound by a substantial amount down to the value given by design‐based constructions. We extend in a few directions.
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 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".