Quasi‐embeddings of Steiner triple systems, or Steiner triple systems of different orders with maximum intersection
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Bibliographic record
Abstract
Abstract In this paper, we present a conjecture that is a common generalization of the Doyen–Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems. Given u , v ≡ 1,3 (mod 6), u < v < 2 u + 1, we ask for the minimum r such that there exists a Steiner triple system $(U,\,{\cal B}),\,|U|=u$ such that some partial system $(U,{\cal B}\,\backslash{\partial})$ can be completed to an STS $(v),\,(V,\,{\cal B}{^\prime})$ , where |∂| = r . In other words, in order to “quasi‐embed” an STS( u ) into an STS( v ), we must remove r blocks from the small system, and this r is the least such with this property. One can also view the quantity ( u ( u − 1)/6) − r as the maximum intersection of an STS( u ) and an STS( v ) with u < v . We conjecture that the necessary minimum r = ( v − u ) (2 u + 1 − v )/6 can be achieved, except when u = 6 t + 1 and v = 6 t + 3, in which case it is r = 3 t for t ≠ 2, or r = 7 when t = 2. Using small examples and recursion, we solve the cases v − u = 2 and 4, asymptotically solve the cases v − u = 6, 8, and 10, and further show for given v − u > 2 that an asymptotic solution exists if solutions exist for a run of consecutive values of u (whose required length is no more than v − u ). Some results are obtained for v close to 2 u + 1 as well. The cases where ≈ 3 u /2 seem to be the hardest. © 2004 Wiley Periodicals, Inc.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 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