Patel 2025 biggest open problem euclidean ramsey theory

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conjecture_3_1: Graham's conjecture as the survey records it: every spherical set is Ramsey, with Graham's prize offer for a proof or counterexample.

conjecture_3_3: Leader, Russell and Walters's conjecture as the survey records it: a set is Ramsey exactly when it is a subset of some finite transitive set.

conjecture_4_1: The survey's conjecture that every 4-point subset of a circle is Ramsey, posed as the next case after triangles.

conjecture_4_3: Leader, Russell and Walters's conjecture as the survey records it: no transcendental kite, a spherical 4-point set that is not subtransitive, is Ramsey.

proposition_3_4: Every subtransitive set is spherical; the survey proves it through the unique smallest closed ball containing a finite transitive set, whose center every isometry of the set fixes.

theorem_2_12: Frankl and Rödl's theorem as the survey records it: every non-degenerate triangle is Ramsey, a statement about sufficiently high dimension and any number of colors.

theorem_2_13: Kříž's theorem that every non-degenerate isosceles trapezoid is Ramsey, with the survey's own proof in Appendix A from Kříž's symmetry-group criterion through anti-drums.

theorem_2_15: Kříž's criterion as the survey states it: a finite configuration with a transitive solvable isometry group, or a transitive one with a solvable subgroup of at most two orbits, is Ramsey.

theorem_2_19: Frankl and Rödl's theorem as the survey records it: every simplex, a set of d + 1 affinely independent points spanning R^d, is Ramsey.

theorem_2_5: Erdős, Graham, Montgomery, Rothschild, Spencer and Straus's necessary condition as the survey records it: a finite set that does not lie on a sphere is not Ramsey.

theorem_2_8: The product theorem of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus as the survey records it: if X and Y are Ramsey, so is X x Y.

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Nikhil Patel, The Biggest Open Problem in Euclidean Ramsey Theory. University of Chicago Mathematics REU 2025 paper. No notice is printed in the file, and the hosting REU page lists the paper and states no copyright, license or terms (https://math.uchicago.edu/~may/REU2025/, read 2026-10-02); the term is unstated.

This expository REU paper, dated August 21, 2025 (15 pp.), surveys the problem of characterizing the finite sets X in R^d that are Ramsey, meaning (Definition 1.2, p. 2) that for every number of colors r there is an n such that every r-coloring of R^n contains a monochromatic congruent copy of X. It assembles the standard toolkit: regular simplices are Ramsey (Proposition 1.4, p. 2), non-spherical sets are not Ramsey (Theorem 2.5, p. 4, Erdős et al.), Ramsey-ness is closed under Cartesian products (Theorem 2.8, p. 5), all non-degenerate triangles (Theorem 2.12, p. 6, Frankl and Rödl) and all non-degenerate isosceles trapezoids (Theorem 2.13, p. 6, Kříž) are Ramsey, Kříž's symmetry-group criterion (Theorem 2.15, p. 6) yields many further examples, and all simplices are Ramsey (Theorem 2.19, p. 8, Frankl and Rödl). Its own contributions are a proof of Theorem 2.13 from Kříž's criterion through anti-drums (Appendix A, pp. 11--14), which it says does not seem to appear in the literature, and a proof that subtransitive sets are spherical (Proposition 3.4, pp. 8--9), which it says is alluded to by Leader, Russell and Walters. Section 3 contrasts Graham's conjecture that all spherical sets are Ramsey (Conjecture 3.1, p. 8, with a prize) with the rival conjecture of Leader, Russell and Walters, "A set is Ramsey if and only if it is subtransitive" (Conjecture 3.3, p. 8). Section 4 poses further problems, among them Conjecture 4.1 (p. 10), that all 4-point subsets of a circle are Ramsey, and Leader, Russell and Walters's Conjecture 4.3 (p. 10), that no transcendental kite is Ramsey. All results concern sufficiently high dimension and arbitrary numbers of colors.

Source: https://math.uchicago.edu/~may/REU2025/REUPapers/Patel.pdf.

Bears on.

• Problem 174: background. The problem asks for a characterization of the Ramsey sets; the paper surveys the known necessary condition (Theorem 2.5), the known classes of Ramsey sets and the two conjectured characterizations (Conjectures 3.1 and 3.3), and proves no new characterization.
• Problem 173: context only. The problem asks about two-colorings of the plane itself; the paper's results on triangles (Theorems 2.12 and 2.13) let the dimension grow with the set and the number of colors, so they do not bear on the problem's question.

Result pages.

• Theorem 2.5 (p. 4): a non-spherical set is not Ramsey.
• Theorem 2.8 (p. 5): the Cartesian product of two Ramsey sets is Ramsey.
• Theorem 2.12 (p. 6): every non-degenerate triangle is Ramsey.
• Theorem 2.13 (p. 6): every non-degenerate isosceles trapezoid is Ramsey, with the paper's proof in Appendix A.
• Theorem 2.15 (p. 6): Kříž's criterion, a transitive solvable isometry group, or a transitive one with a solvable subgroup of at most two orbits, makes a set Ramsey.
• Theorem 2.19 (p. 8): every simplex is Ramsey.
• Conjecture 3.1 (p. 8): Graham's conjecture that every spherical set is Ramsey.
• Conjecture 3.3 (p. 8): Leader, Russell and Walters's conjecture that a set is Ramsey exactly when it is subtransitive.
• Proposition 3.4 (pp. 8--9): every subtransitive set is spherical.
• Conjecture 4.1 (p. 10): every 4-point subset of a circle is Ramsey.
• Conjecture 4.3 (p. 10): no transcendental kite is Ramsey.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.