Canonical Ramsey Numbers of Sparse Graphs
Bibliographic record
Abstract
Abstract. The canonical Ramsey theorem of Erdős and Rado implies that for any graph [Formula: see text], any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph [Formula: see text] contains a monochromatic, lexicographic, or rainbow copy of [Formula: see text]. The least such [Formula: see text] is called the Erdős–Rado number of [Formula: see text], denoted by [Formula: see text]. Erdős–Rado numbers of cliques have received considerable attention, and in this paper we extend this line of research by studying Erdős–Rado numbers of sparse graphs. For example, we prove that if [Formula: see text] has bounded degree, then [Formula: see text] is polynomial in [Formula: see text] if [Formula: see text] is bipartite but exponential in general. We also study the closely related problem of constrained Ramsey numbers. For a given tree [Formula: see text] and given path [Formula: see text], we study the minimum [Formula: see text] such that every edge-coloring of [Formula: see text] contains a monochromatic copy of [Formula: see text] or a rainbow copy of [Formula: see text]. We prove a nearly optimal upper bound for this problem, which differs from the best known lower bound by a function of inverse Ackermann type.
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.
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 | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.011 | 0.001 |
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".