Graham 2010 open problems euclidean ramsey theory

../

conjecture_1: The survey's Conjecture 1, credited to Erdős, Graham, Montgomery, Rothschild, Spencer and Straus: every non-equilateral triangle has a monochromatic congruent copy in every two-coloring of the plane.

conjecture_2: The survey's Conjecture 2, that each triangle has a three-coloring of the plane with no monochromatic congruent copy, and the authors' hexagonal three-coloring avoiding the degenerate triangle with sides a, a and 2a.

conjecture_3: The survey's Conjecture 3, with a prize: every spherical set is Ramsey, which with the reported converse would make the Ramsey sets exactly the spherical ones.

conjecture_4: The survey's Conjecture 4, with a prize and posed as weaker than Conjecture 3: every four-point subset of a circle is Ramsey.

counterexample_p4: The authors' negative answer to Chilakamarri's question whether every bipartite graph that is not a planar unit distance graph contains K_{2,3}: the five-dimensional cube with its opposite vertices joined.

theorem_1: Kříž's theorem as the survey states it: a set in Euclidean space with a transitive group of isometries that has a solvable subgroup with at most two orbits is Ramsey.

theorem_2: Kříž's theorem as the survey states it: the four vertices of any trapezoid form a Ramsey set.

unit_distance_survey_p3: The bounds the survey reports from other authors in pp. 3 to 4: the chromatic number of the plane lies between 4 and 7, and is at least 5 if every set is measurable; small-dimensional and rational values follow, and Croft's planar set avoiding distance one has density above 0.2294.

---

Ron Graham, 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.

Graham and Tressler collect open problems in Euclidean Ramsey theory. Section 1 (pp. 1--2) states Conjecture 1 (every non-equilateral triangle is 2-Ramsey in the plane) and Conjecture 2 (every triangle admits a 3-coloring of the plane with no monochromatic copy), reporting that Jelínek, Kynčl, Stolař and Valla found infinitely many 2-colorings avoiding a given equilateral triangle, where the alternating-strip coloring had been conjectured essentially unique, Shader's right-triangle case, and the authors' own 3-coloring by half-open hexagons of diameter 2a avoiding the degenerate (a,a,2a) triangle. Section 2 (pp. 2--3) states Kříž's theorem on sets with a transitive group of isometries having a solvable subgroup with at most two orbits and Kříž's trapezoid theorem, reports that every Ramsey set is spherical, and poses the prize conjectures that every spherical set is Ramsey and that every 4-point subset of a circle is Ramsey. Section 3 (pp. 3--4) surveys unit distance graphs: $4\le\chi(\mathbb{E}^2)\le7$ with the Moser spindle and the hexagonal 7-coloring, Falconer's $\chi(\mathbb{E}^2)\ge5$ when all sets are assumed measurable, $6\le\chi(\mathbb{E}^3)\le15$, $7\le\chi(\mathbb{E}^4)\le49$, the polychromatic bounds $4\le\chi_p(\mathbb{E}^2)\le6$, the rational values $\chi(\mathbb{Q}^2)=\chi(\mathbb{Q}^3)=2$ and $\chi(\mathbb{Q}^4)=4$, O'Donnell's 4-chromatic unit distance graphs of arbitrary girth, the authors' sketched hypercube counterexample to Chilakamarri's question whether every bipartite graph that is not a unit distance graph contains $K_{2,3}$, and Croft's planar set of density above 0.2294 avoiding distance one. Section 4 (pp. 4--5) reports the odd-distance graph of the plane having chromatic number at least 5. The (a,a,2a) coloring is the authors' own, and the hypercube counterexample is sketched without attribution; the other results are credited to the works the survey cites. The survey does not ask whether the complement of a planar set avoiding distance one must contain the vertices of a unit square.

Source: https://mathweb.ucsd.edu/~ronspubs/10_14_euclid.pdf. The copy read for this card is the authors' survey preprint, which prints no copyright or license line, and the author's homepage (mathweb.ucsd.edu/~ronspubs) could not be read on 2026-10-02; the publisher's edition is not the edition read; the term is unstated.

Bears on.

• #173: Conjecture 1 would confine the triangles a two-coloring of the plane can avoid to equilateral ones; the paper poses it and proves nothing toward the problem.
• #174: Kříž's Theorem 1 and Theorem 2, as reported, give sufficient conditions for a set to be Ramsey; Conjecture 3, with the reported theorem that Ramsey sets are spherical, would characterise the Ramsey sets as the spherical ones, and Conjecture 4 is its four-point case. The paper poses both as conjectures.
• #508: the unit distance survey reports the bounds $4\le\chi(\mathbb{E}^2)\le7$ known when it was written, and Falconer's bound 5 under the assumption that every set is measurable.
• #704: the same page reports bounds for $\chi(\mathbb{E}^3)$ and $\chi(\mathbb{E}^4)$ only, which say nothing about growth in $n$.
• #705: the same page reports, without proof, O'Donnell's 4-chromatic unit distance graphs of arbitrary girth, which answer the question no.
• #232: the same page reports Croft's measurable planar set avoiding distance one with density more than 0.2294, a lower bound for the problem's $m_1$.
• #214: no result in the survey addresses the question; the survey treats Ramsey sets, chromatic numbers and dense sets avoiding distance one, but not the unit square in the complement of such a set.

Results.

• Conjecture 1 (p. 1): every non-equilateral triangle has a monochromatic congruent copy in every two-coloring of the plane.
• Conjecture 2 (p. 1): every triangle is avoided by some three-coloring of the plane; with the authors' hexagonal three-coloring avoiding the (a,a,2a) triangle (p. 2).
• Theorem 1 (p. 2, Kříž): a set with a transitive group of isometries having a solvable subgroup with at most two orbits is Ramsey.
• Theorem 2 (p. 2, Kříž): the vertex set of a trapezoid is Ramsey.
• Conjecture 3 (p. 3, with a prize): every spherical set is Ramsey.
• Conjecture 4 (p. 3, with a prize): every 4-point subset of a circle is Ramsey.
• Unit distance survey (pp. 3--4): the reported bounds on the chromatic numbers of the plane, of 3- and 4-space and of rational space, the polychromatic number, high-girth 4-chromatic graphs, and dense measurable sets avoiding distance one.
• Counterexample (p. 4): $Q_5$ with its 16 space diagonals is bipartite, has no $K_{2,3}$ and is not a unit distance graph in the plane.

Page numbers are those of the authors' preprint, the edition read.

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