Small forbidden configurations V: Exact bounds for 4 × 2 cases
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Bibliographic record
Abstract
The present paper continues the work begun by Anstee, Ferguson, Griggs, Kamoosi and Sali on small forbidden configurations. We define a matrix to be simple if it is a (0, 1)-matrix with no repeated columns. Let F be a k × (0, 1)-matrix (the forbidden configuration). Assume A is an m × n simple matrix which has no submatrix which is a row and column permutation of F . We define forb ( m, F ) as the largest n , which would depend on m and F , so that such an A exists.Define F abcd as the ( a + b + c + d ) × 2 matrix consisting of a rows of [11], b rows of [10], c rows of [01] and d rows of [00]. With the exception of F 2110 , we compute forb ( m; F abcd ) for all 4 × 2 F abcd . A number of cases follow easily from previous results and general observations. A number follow by clever inductions based on a single column such as forb ( m; F 1111 ) = 4 m − 4 and forb ( m; F 1210 ) = forb ( m; F 1201 ) = forb ( m; F 0310 ) = ( 2 m )+ m + 2 (proofs are different). A different idea proves forb ( m; F 0220 ) = ( 2 m ) + 2 m − 1 with the forbidden configuration being related to a result of Kleitman. Our results suggest that determining forb ( m; F 2110 ) is heavily related to designs and we offer some constructions of matrices avoiding F 2110 using existing designs.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it