Corrádi and Hajnal's Theorem for Sparse Random Graphs
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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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Full frame distilled prediction
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 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)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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