Towards a Weighted Version of the Hajnal–Szemerédi Theorem
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Bibliographic record
Abstract
For a positive integer r ≥ 2, a K r -factor of a graph is a collection vertex-disjoint copies of K r which covers all the vertices of the given graph. The celebrated theorem of Hajnal and Szemerédi asserts that every graph on n vertices with minimum degree at least $(1-\frac{1}{r})n contains a K r -factor. In this note, we propose investigating the relation between minimum degree and existence of perfect K r -packing for edge-weighted graphs. The main question we study is the following. Suppose that a positive integer r ≥ 2 and a real t ∈ [0, 1] is given. What is the minimum weighted degree of K n that guarantees the existence of a K r -factor such that every factor has total edge weight at least $$t\binom{r}{2}$?$ We provide some lower and upper bounds and make a conjecture on the asymptotics of the threshold as n goes to infinity.
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.008 |
| Research integrity | 0.001 | 0.002 |
| 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