Integer and fractional packings in dense 3‐uniform hypergraphs
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Bibliographic record
Abstract
Abstract Let 𝒥 0 be any fixed 3‐uniform hypergraph. For a 3‐uniform hypergraph ℋ︁ we define ν (ℋ︁) to be the maximum size of a set of pairwise triple‐disjoint copies of 𝒥 0 in ℋ︁. We say a function ψ from the set of copies of 𝒥 0 in ℋ︁ to [0, 1] is a fractional 𝒥 0 ‐ packing of ℋ︁ if ∑ 𝒥∋ e ψ(𝒥) ≤ 1 for every triple e of ℋ︁. Then ν (ℋ︁) is defined to be the maximum value of ∑ ψ(𝒥) over all fractional 𝒥 0 ‐packings ψ of ℋ︁. We show that ν (ℋ︁) − ν (ℋ︁) = o (| V (ℋ︁)| 3 ) for all 3‐uniform hypergraphs ℋ︁. This extends the analogous result for graphs, proved by Haxell and Rödl (2001), and requires a significant amount of new theory about regularity of 3‐uniform hypergraphs. In particular, we prove a result that we call the Extension Theorem. This states that if a k ‐partite 3‐uniform hypergraph is regular [in the sense of the hypergraph regularity lemma of Frankl and Rödl (2002)], then almost every triple is in about the same number of copies of K (the complete 3‐uniform hypergraph with k vertices). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 248–310, 2003
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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 it