Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Source. Published pp. 345, 347, 350, 351, 356, 358, 360 and 362 (published scan). These are questions and context from 1973, not a current status census.
The paper asks whether sphericality suffices for a finite configuration to be Ramsey. Its necessity proof is now reconstructed at theorem_13. The broader characterization question belongs to Problem 174, where any present-day status needs separately dated literature evidence.
The source's example of an undecided Ramsey configuration on p. 360 is an obtuse triangle. Every noncollinear triangle is an affinely independent simplex, so the later complete Frankl–Rödl simplex proof and its ordinary Ramsey consequence resolve that historical question. Collinear triples fall under the nonspherical obstruction instead. The original Theorem 8's fixed two-color conclusion is a different quantifier statement from being Ramsey for every finite number of colors.
The original product theorem is a cited input in theorem_2_1. Its finite-witness pattern argument is now fully compiled at theorem_20. The stronger density product proof from 1990 and the recent pyramid methods are recorded as distinct arguments, not attributed back to the 1973 paper.
The source also discusses planar two-color triangle questions, posed as the conjecture on p. 347, dimensions for forcing a square, a uniform color bound for collinear triples, and whether -Ramsey sets can be characterized by containment in concentric spheres. The few-color product theorem is explicitly for finite factors. These historical questions are not labeled currently open merely because the original paper leaves them unanswered.
Two compressed source extensions are fully supplied in this compilation: the finite-witness argument for infinite nonspherical or concentric-sphere obstructions, and the exact placement on every sufficiently large sphere in Theorem 24. By contrast the general “nice” surface embedding comment on p. 351 remains an external thesis reference, and the negative seven-color part of Theorem 5 remains an external construction.
Bears on. #174: the question whether sphericality suffices (pp. 350 and 360) asks whether the necessary condition of Theorem 13 characterizes the Ramsey sets. #173: the planar triangle conjecture of p. 347 implies the problem's statement, as its page shows.