Quasi‐embeddings of Steiner triple systems, or Steiner triple systems of different orders with maximum intersection
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. Givenu,v≡ 1,3 (mod 6),u<v< 2u + 1, we ask for the minimumrsuch 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 removerblocks from the small system, and thisris the least such with this property. One can also view the quantity (u(u− 1)/6) −ras the maximum intersection of an STS(u) and an STS(v) withu<v. We conjecture that the necessary minimumr = (v−u) (2u + 1 −v)/6 can be achieved, except whenu = 6t + 1 andv= 6t + 3, in which case it isr = 3tfort≠ 2, orr = 7 whent = 2. Using small examples and recursion, we solve the casesv−u = 2 and 4, asymptotically solve the casesv−u = 6, 8, and 10, and further show for givenv−u> 2 that an asymptotic solution exists if solutions exist for a run of consecutive values ofu(whose required length is no more thanv−u). Some results are obtained forvclose to 2u + 1 as well. The cases where ≈ 3u/2 seem to be the hardest. © 2004 Wiley Periodicals, Inc.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.010 | 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".