The $t$-Stability Number of a Random Graph
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Bibliographic record
Abstract
Given a graph $G = (V,E)$, a vertex subset $S \subseteq V$ is called $t$-stable (or $t$-dependent) if the subgraph $G[S]$ induced on $S$ has maximum degree at most $t$. The $t$-stability number $\alpha_t(G)$ of $G$ is the maximum order of a $t$-stable set in $G$. The theme of this paper is the typical values that this parameter takes on a random graph on $n$ vertices and edge probability equal to $p$. For any fixed $0 < p < 1$ and fixed non-negative integer $t$, we show that, with probability tending to $1$ as $n\to\infty$, the $t$-stability number takes on at most two values which we identify as functions of $t$, $p$ and $n$. The main tool we use is an asymptotic expression for the expected number of $t$-stable sets of order $k$. We derive this expression by performing a precise count of the number of graphs on $k$ vertices that have maximum degree at most $t$.
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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 | 0.005 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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