Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Statement
Definitions (p. 286). A finite configuration in Euclidean space is transitive if it has a transitive group of symmetries, and subtransitive if it is a subset of a transitive configuration. Ramsey is as on the Theorem 11.2.5 page.
Conjecture 11.2.14 (p. 286), credited to Leader, Russell and Walters (2012). Every Ramsey set is subtransitive.
The chapter adds that the same authors showed, in a 2011 paper, that almost all 4-point subsets of a unit circle are not subtransitive, so that whether the 4-point subsets of a circle are Ramsey separates this conjecture from Conjecture 11.2.13.
Scope
This is a conjecture restated from its authors, not a result of the chapter. The corpus records the conjecture at its source, Leader, Russell and Walters.
Source. R. L. Graham, Euclidean Ramsey theory, Chapter 11 of J. E. Goodman, J. O'Rourke and C. D. Tóth (eds.), Handbook of Discrete and Computational Geometry, 3rd edition, CRC Press, Boca Raton, FL, 2017; the definitions, the conjecture and the remark after it on p. 286. Pages are those printed on the edition named on the source card.
Read depth. Claims checked: the definitions, the statement and the remark were read on the printed page.
Bears on
- Problem 174: the conjecture states a necessary condition for a set to be Ramsey; with the converse, which Leader, Russell and Walters also propose and the chapter does not state, it would characterize the Ramsey sets. The problem page records the later work that bears on it.