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 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 | 0.000 | 0.000 |
| 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".