Constructions of Turán systems that are tight up to a multiplicative constant
Bibliographic record
Abstract
For positive integers n ⩾ s > r , the Turán function T ( n , s , r ) is the smallest size of an r -graph with n vertices such that every set of s vertices contains at least one edge. Also, define the Turán density t ( s , r ) as the limit of T ( n , s , r ) / ( n r ) as n → ∞ . The question of estimating these parameters received a lot of attention after it was first raised by Turán in 1941. A trivial lower bound is t ( s , r ) ⩾ 1 / ( s s − r ) . In the 1990s, de Caen conjectured that r ⋅ t ( r + 1 , r ) → ∞ as r → ∞ and offered 500 Canadian dollars for resolving this question. We disprove this conjecture by showing more strongly that for every integer R ⩾ 1 there is μ R (in fact, μ R can be taken to grow as ( 1 + o ( 1 ) ) R ln R ) such that t ( r + R , r ) ⩽ ( μ R + o ( 1 ) ) / ( r + R R ) as r → ∞ , that is, the trivial lower bound is tight for every R up to a multiplicative constant μ R .
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.002 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.002 | 0.006 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.011 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".