Bibliographic record
Abstract
This thesis consists of two papers within two different areas of combinatorics.Ramsey theory is a classic topic in graph theory, and Paper A deals with two of its most fundamental problems: to compute Ramsey numbers and to characterise critical graphs.More precisely, we study generalised Ramsey numbers for two sets Γ 1 and Γ 2 of cycles.We determine, in particular, all generalised Ramsey numbers R(Γ 1 , Γ 2 ) such that Γ 1 or Γ 2 contains a cycle of length at most 6, or the shortest cycle in each set is even.This generalises previous results of Erdős, Faudree, Rosta, Rousseau, and Schelp.Furthermore, we give a conjecture for the general case.We also characterise many (Γ 1 , Γ 2 )critical graphs.As special cases, we obtain complete characterisations of all (C n , C 3 )-critical graphs for n ≥ 5, and all (C n , C 5 )-critical graphs for n ≥ 6.In Paper B, we study the combinatorics of certain partially ordered sets.These posets are unions of conjugacy classes of involutions in the symmetric group S n , with the order induced by the Bruhat order on S n .We obtain a complete characterisation of the posets that are graded.In particular, we prove that the set of involutions with exactly one fixed point is graded, which settles a conjecture of Hultman in the affirmative.When the posets are graded, we give their rank functions.We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, recently proved by Can, Cherniavsky, and Twelbeck.iii
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 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".