An Extension of the Erd\H{o}s-Ko-Rado Theorem to uniform set partitions
Bibliographic record
Abstract
A $(k,\\ell)$-partition is a set partition which has $\\ell$ blocks each of\nsize $k$. Two uniform set partitions $P$ and $Q$ are said to be partially\n$t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such\nthat $\\left| P_{i} \\cap Q_{j} \\right|\\geq t$. In this paper we prove a version\nof the Erd\\H{o}s-Ko-Rado theorem for partially $2$-intersecting\n$(k,\\ell)$-partitions. In particular, we show for $\\ell$ sufficiently large,\nthe set of all $(k,\\ell)$-partitions in which a block contains a fixed pair is\nthe largest set of 2-partially intersecting $(k,\\ell)$-partitions. For for\n$k=3$, we show this result holds for all $\\ell$.\n
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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 | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| 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".