Decompositions of edge-colored infinite complete graphs into monochromatic paths
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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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".