Why this work is in the frame
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Bibliographic record
Abstract
Mantel's Theorem from 1907 is one of the oldest results in graph theory: every simple $n$-vertex graph with more than $\frac{1}{4}n^2$ edges contains a triangle. The theorem has been generalized in many different ways, including other subgraphs, minimum degree conditions, etc. This article deals with a generalization to edge-colored multigraphs, which can be viewed as a union of simple graphs, each corresponding to an edge-color class. The case of two colors is the same as the original setting: Diwan and Mubayi proved that any two graphs $G_1$ and $G_2$ on the same set of $n$ vertices, each containing more than $\frac{1}{4}n^2$ edges, give rise to a triangle with one edge from $G_1$ and two edges from $G_2$. The situation is however different for three colors. Fix $\tau=\frac{4-\sqrt{7}}{9}$ and split the $n$ vertices into three sets $A$, $B$ and $C$, such that $|B|=|C|=\lfloor\tau n\rfloor$ and $|A|=n-|B|-|C|$. The graph $G_1$ contains all edges inside $A$ and inside $B$, the graph $G_2$ contains all edges inside $A$ and inside $C$, and the graph $G_3$ contains all edges between $A$ and $B\cup C$ and inside $B\cup C$. It is easy to check that there is no triangle with one edge from $G_1$, one from $G_2$ and one from $G_3$; each of the graphs has about $\frac{1+\tau^2}{4}n^2=\frac{26-2\sqrt{7}}{81}n^2\approx 0.25566n^2$ edges. The main result of the article is that this construction is optimal: any three graphs $G_1$, $G_2$ and $G_3$ on the same set of $n$ vertices, each containing more than $\frac{1+\tau^2}{4}n^2$ edges, give rise to a triangle with one edge from each of the graphs $G_1$, $G_2$ and $G_3$. A computer-assisted proof of the same bound has been found by Culver, Lidický, Pfender and Volec.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| 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