Linear independence over naturally-ordered semirings with applications to dimension arguments in extremal combinatorics
Bibliographic record
Abstract
A family of subsets F ⊆ P ( { 1 , 2 , … , n } ) has the disparate union property if any two disjoint subfamilies F 1 , F 2 ⊆ F (not both empty) have distinct unions ⋃ F 1 ≠ ⋃ F 2 ; what is the maximal size of a family with the disparate union property? Is there a simple and efficiently computable characterization of size-maximal families? This paper highlights a class of partially-ordered semirings—difference ordered semirings with a multiplicatively absorbing element—and shows it is common and easily constructed. We prove that a suitably modified definition of linear independence for semimodules over such semirings enjoys the same maximality property as for vector spaces, and can furthermore be efficiently detected by the bideterminant. These properties allow us to extend dimension argument in extremal combinatorics and provide simple and direct solutions to the problems above.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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".