Decompositions of edge-colored infinite complete graphs into monochromatic paths
Why this work is in the frame
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Bibliographic record
Abstract
An r -edge coloring of a graph or hypergraph G = ( V , E ) is a map c : E → { 0 , … , r − 1 } . Extending results of Rado and answering questions of Rado, Gyárfás and Sárközy we prove that • the vertex set of every r -edge colored countably infinite complete k -uniform hypergraph can be partitioned into r monochromatic tight paths with distinct colors (a tight path in a k -uniform hypergraph is a sequence of distinct vertices such that every set of k consecutive vertices forms an edge); • for all natural numbers r and k there is a natural number M such that the vertex set of every r -edge colored countably infinite complete graph can be partitioned into M monochromatic k th powers of paths apart from a finite set (a k th power of a path is a sequence v 0 , v 1 , … of distinct vertices such that 1 ⩽ | i − j | ⩽ k implies that v i v j is an edge); • the vertex set of every 2 -edge colored countably infinite complete graph can be partitioned into 4 monochromatic squares of paths, but not necessarily into 3 ; • the vertex set of every 2 -edge colored complete graph on ω 1 can be partitioned into 2 monochromatic paths with distinct colors.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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 | 0.000 |
| 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