Siblings of an ℵ0-categorical relational structure
Bibliographic record
Abstract
A sibling of a relational structure $R$ is any structure $S$ which can be embedded into $R$ and, vice versa, such that $R$ can be embedded into $S$. Let $\operatorname{sib}(R)$ be the number of siblings of $R$, these siblings being counted up to isomorphism. Thomassé conjectured that for countable relational structures made of at most countably many relations, $\operatorname{sib}(R)$ is either one, countably infinite, or the size of the continuum; but even showing the special case $\operatorname{sib}(R)1$ is one or infinite is unsettled when $R$ is a countable tree. We prove that if $R$ is countable and $\aleph_{0}$-categorical, then indeed $\operatorname{sib}(R)$ is one or infinite. Furthermore, $\operatorname{sib}(R)$ is one if and only if $R$ is finitely partitionable in the sense of Hodkinson and Macpherson [14]. The key tools in our proof are the notion of monomorphic decomposition of a relational structure introduced in [35] and studied further in [23], [24], and a result of Frasnay [11].
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".