The First<i>k</i>-Regular Subgraph is Large
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Bibliographic record
Abstract
Let $(G_m)_{0\le m\le \binom{n}{2}}$ be the random graph process starting from the empty graph on vertex set [ n ] and with a random edge added in each step. Let m k denote the minimum integer such that G m k contains a k -regular subgraph. We prove that for all sufficiently large k , there exist two constants ε k ≥ σ k > 0, with ε k → 0 as k → ∞, such that asymptotically almost surely any k -regular subgraph of G m k has size between (1 − ε k )| ${\mathcal C}_k$ | and (1 − σ k )| ${\mathcal C}_k$ |, where ${\mathcal C}_k$ denotes the k -core of G m k .
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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 | 0.004 | 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 | 0.003 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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 it