A rainbow version of Mantel's Theorem
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.
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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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.026 | 0.006 |
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".