Typical Structure of Hereditary Graph Families. II. Exotic Examples
Bibliographic record
Abstract
ABSTRACT A graph is ‐free if it does not contain an induced subgraph isomorphic to . The study of the typical structure of ‐free graphs was initiated by Erdős, Kleitman, and Rothschild (1976), who have shown that almost all ‐free graphs are bipartite. Since then the typical structure of ‐free graphs has been determined for several families of graphs , including complete graphs, trees, and cycles. Recently, Reed and Scott proposed a conjectural description of the typical structure of ‐free graphs for all graphs , which extends all previously known results in the area. We construct an infinite family of graphs for which the Reed–Scott conjecture fails, and use the methods we developed in the prequel paper Norin and Yuditsky (2024) to describe the typical structure of ‐free graphs for graphs in this family. Using similar techniques, we construct an infinite family of graphs for which the maximum size of a homogenous set in a typical ‐free graph is sublinear in the number of vertices, answering a question of Loebl et al. (2010) and Kang et al. (2014).
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".