Bibliographic record
Abstract
We explore an extremal hypergraph problem for which both the vertices and edges are ordered. Given a hypergraph $F$ (not necessarily simple), we consider how many edges a simple hypergraph (no repeated edges) on $m$ vertices can have while forbidding $F$ as a subhypergraph where both hypergraphs have fixed vertex and edge orderings. A hypergraph of $n$ edges on $m$ vertices can be encoded as an $m\times n$ (0,1)-matrix. We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given a (0,1)-matrix $F$, we define ${\hbox{fs}}(m,F)$ as the maximum, over all simple matrices $A$ which do not have $F$ as a submatrix, of the number of columns in $A$. The row and column order matter. It is known that if $F$ is $k\times \ell$ then ${\hbox{fs}}(m,F)$ is $O(m^{2k-1-\epsilon})$ where $\epsilon=(k-1)/(13\log_2 \ell)$. Anstee, Frankl, Füredi and Pach have conjectured that if $F$ is $k$-rowed, then ${\hbox{fs}}(m,F)$ is $O(m^k)$. We show ${\hbox{fs}}(m,F)$ is $O(m^2)$ for $F= \left[{1\,0\,1\,0\,1\atop 0\,1\,0\,1\,0}\cdots\right]$ and for $F= \left[{1\,0\,1\,0\,1\atop 1\,0\,1\,0\,1}\cdots\right]$. The proofs use a type of amortized analysis. We also give some constructions.
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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 | 0.003 | 0.019 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.006 | 0.014 |
| Open science | 0.005 | 0.008 |
| Research integrity | 0.003 | 0.011 |
| Insufficient payload (model declined to judge) | 0.027 | 0.005 |
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".