Why this work is in the frame
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Bibliographic record
Abstract
Monochromatic sums and products, Discrete Analysis 2016:5, 48pp. An old and still unsolved problem in Ramsey theory asks whether if the positive integers are coloured with finitely many colours, then there are positive integers $x$ and $y$ such that $x, y, x+y$ and $xy$ all have the same colour. In fact, it is not even known whether it is always possible to find $x$ and $y$ such that $x+y$ and $xy$ have the same colour. This paper is about the corresponding question when $\mathbb{N}$ is replaced by a finite field $\mathbb{F}_p$, and gives a positive answer. More than that, it proves that a positive fraction of the quadruples $(x,y,x+y,xy)$ are monochromatic. The result is interesting for several reasons. One is that the standard tools of Ramsey theory appear to be hopelessly inadequate when they are applied to questions that mix addition and multiplication, so the fact that the authors have obtained a positive result of this kind is surprising and may well have further ramifications. Another is that several people have tried, without much success, to apply techniques from additive combinatorics to colouring problems. The techniques work well for many density problems, from which one can of course deduce colouring results. Until now the challenge has been to get them to work for colouring problems when the corresponding density statements are false, as is the case here (since the set of numbers between $p/3$ and $2p/3$ does not even contain a triple of the form $(x,y,x+y)$). A third reason, which is almost implied by the previous two, is that the paper introduces some striking new techniques. One of these techniques is to "smooth" the colouring in a way that converts the count of quadruples $(x,y,x+y,xy)$ into a count of purely linear configurations, thereby making the problem more amenable to conventional Ramsey-theoretic techniques. It also uses deep character sum estimates from number theory. For all these reasons, the paper will repay careful study by those who work in additive combinatorics.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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