On the ($k + \text{2}, k$)-problem of Brown, Erdős, and Sós for $k =$ 5,6,7
Bibliographic record
Abstract
Abstract Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an n -vertex r -uniform hypergraph containing no subgraph with k edges and at most s vertices. Brown, Erdős, and Sós [ New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan 1971) , pp. 53–63, Academic Press 1973] conjectured that the limit $\lim _{n\rightarrow \infty }n^{-2}f^{(3)}(n;k+2,k)$ exists for all k . The value of the limit was previously determined for $k=2$ in the original paper of Brown, Erdős, and Sós, for $k=3$ by Glock [ Bull. Lond. Math. Soc., 51 (2019) 230–236] and for $k=4$ by Glock, Joos, Kim, Kühn, Lichev, and Pikhurko [ Proc. Amer. Math. Soc., Series B , 11 (2024) 173–186] while Delcourt and Postle [ Proc. Amer. Math. Soc. , 152 (2024), 1881–1891] proved the conjecture (without determining the limiting value). In this article, we determine the value of the limit in the Brown–Erdős–Sós problem for $k\in \{5,6,7\}$ . More generally, we obtain the value of $\lim _{n\rightarrow \infty }n^{-2}f^{(r)}(n;rk-2k+2,k)$ for all $r\geqslant 3$ and $k\in \{5,6,7\}$ . In addition, by combining these new values with recent results of Bennett, Cushman, and Dudek [arxiv:2309.00182, 2023] we obtain new asymptotic values for several generalized Ramsey numbers.
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How this classification was reachedexpand
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 | 0.003 |
| 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 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".