Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Source. Conjecture 3 on p. 3 of Ron Graham and Eric Tressler, Open problems in Euclidean Ramsey theory, in A. Soifer (ed.), Ramsey Theory: Yesterday, Today, and Tomorrow, Progress in Mathematics, Birkhäuser (2011), 115--120, doi:10.1007/978-0-8176-8092-3_7. Page numbers here are those of the authors' preprint, the edition read, as identified on the source card.
Statement
Setting (pp. 1--2). Ramsey sets are finite (p. 1), and a set is spherical when it lies on the surface of a sphere in some dimension (p. 2). The paper reports (p. 2, its reference [8]: Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, Euclidean Ramsey Theory I) that every Ramsey set is spherical.
Conjecture 3 (p. 3, quoted). "($1000). Every spherical set is Ramsey."
The paper introduces it with the remark that the Ramsey sets may turn out to be very easy to describe (p. 2), and does not prove or refute it.
Read depth. Claims checked: the statement and its setting were read on pp. 1--3 of the preprint.
Bears on
- Problem 174: together with the reported theorem that every Ramsey set is spherical, Conjecture 3 would characterise the Ramsey sets as the finite spherical sets. The paper poses it as a conjecture.