Corrádi and Hajnal's Theorem for Sparse Random Graphs
Bibliographic record
Abstract
In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if p ( n ) ≫ (log n / n ) 1/2 , then asymptotically almost surely every subgraph of G ( n , p ) with minimum degree at least (2/3 + o (1)) np contains a triangle packing that covers all but at most O ( p −2 ) vertices. Moreover, the assumption on p is optimal up to the (log n ) 1/2 factor and the presence of the set of O ( p −2 ) uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which says that if one imposes certain natural regularity conditions on all three pairs in a balanced 3-partite graph, then this graph contains a perfect triangle packing.
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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 | 0.007 | 0.050 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.003 | 0.007 |
| Scholarly communication | 0.003 | 0.012 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.003 | 0.006 |
| Insufficient payload (model declined to judge) | 0.009 | 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".