A sharp lower bound on the number of hyperedges in a friendship 3-hypergraph.
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Bibliographic record
Abstract
Let (X,B) be a set system in which B is a set of 3-subsets of X. Then (X,B) is a friendship 3hypergraph if it satisfies the following property: for all distinct elements u, v, w ∈ X, there exists a unique fourth element x ∈ X such that {u, v, x}, {u,w, x}, {v, w, x} ∈ B. The element x is called the completion of u, v, w and we say u, v, w is completed by x. If a friendship 3-hypergraph contains an element f ∈ X such that {f, u, v} ∈ B for all u, v ∈ X {f}, then the friendship 3-hypergraph is called a universal friend 3-hypergraph and the element f is called a universal friend of the hypergraph. In this note, we show that if (X,B) is a friendship 3-hypergraph with |X| = n, then |B| ≥ d2(n − 1)(n − 2)/3e. In addition, we show that this bound is met if and only if (X,B) is a universal friend 3-hypergraph.
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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 | 0.000 | 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) | 0.001 | 0.001 |
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