Regularity, uniformity, and quasirandomness
Why this work is in the frame
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Bibliographic record
Abstract
Rodl et al. extend a powerful tool, the regularity lemma, from graphs to hypergraphs. Graph theory is the appropriate language for discussing binary relations on objects. Results in graph theory have numerous applications in biology, chemistry, computer science, and physics. In cases of multiple relations, instead of binary relations more general structures known as hypergraphs are the right tools. However, it turns out that because of their extremely complex structure, hypergraphs are very difficult to deal with. As with number theory, there are questions about hypergraphs that are easy to state but very difficult to answer. In this issue of PNAS, Rodl et al. (1) extend a powerful tool, the regularity lemma, from graphs to hypergraphs. Contrary to the general terminology, in extremal graph theory regularity is a measure of randomness. Random graphs are easy to work with, especially when one wants to estimate the (expected) number of small subgraphs. In complex structures, like in dense graphs, one can substitute randomness with weaker but still useful properties. The motivation behind graph regularity is to arrange the vertices of a graph in such a way that the graph becomes similar to the union of a few random graphs, and then one can apply standard counting methods from probability theory. In order to define hypergraph regularity, one has to introduce somehow complicated and technical notations. However, even without these notations we can formulate the most important consequence of the so-called hypergraph regularity method. The method, which is the combination of the hypergraph regularity lemma and a counting lemma is described by Rodl et al. (1). Similar results with the same consequences have been obtained independently by Gowers (2). Inspired by the methods of refs. 1 and 2, very recently Tao (T. Tao, personal communication) gave another proof of the main results. …
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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