Euclidean Ramsey Theorems I

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brick_corollaries: Derives Ramsey bricks, their subsets and regular-simplex dual configurations from product closure.

compactness: Derives finite witnesses for finite color constraints, including monochromatic and few-color copies.

conjecture_p347: The paper's unnumbered conjecture that R(T,2,2) holds for every non-equilateral triangle T, and that a two-coloring of the plane missing the equilateral triangle of one side has every other equilateral triangle.

corollary_10: Deduces the planar two-color Ramsey property for a 30°–60° right triangle from Theorem 9's three forced scales.

corollary_18: Constructs at most (2k)^k shell colors when the affine-relation coefficient ratios are algebraically independent.

corollary_19: Proves the algebraic and transcendental irrational-ratio cases with signed half-interval errors.

definitions: Fixes color, dimension and sphere conventions and proves the elementary closure and regular-simplex facts.

external_inputs: Separates the cited combinatorial and compactness inputs from the reconstructed field and geometry proofs.

finite_sphere_obstruction: Proves the finite obstruction needed for the source extensions to infinite configurations.

historical_questions: Separates the original open questions and source limits from later compiled results.

lemma_14: Characterizes nonspherical finite sets by an affine relation with nonzero squared-norm discrepancy.

lemma_15: Derives the scalar coloring obstruction needed for the spherical necessity theorem.

lemma_26: Proves the linear consistency criterion for a common point equidistant from prescribed pairs.

lemma_27: Uses the complete field theorem to forbid simultaneous color agreement for pairs without a common bisector.

nonobtuse_five_point_obstruction: Expands the source five-point example and distinguishes its affine dimension from a nondegenerate simplex.

theorem_11: Expands the sphere-band argument proving that every two-coloring of three-space contains a monochromatic copy of three collinear unit-spaced points with a unit perpendicular edge at one endpoint.

theorem_12: Gives exact radial shell colorings in every dimension that avoid three, four, and six collinear unit-spaced points with four, three, and two colors.

theorem_13: Combines the complete affine-relation and field-coloring proofs into the dimension-uniform spherical obstruction.

theorem_16: Proves the full field-extension argument excluding prescribed nonzero linear sums of same-color differences.

theorem_17: Proves that any finite coloring of the rationals admits two same-color differences whose product is one.

theorem_20: Proves product closure with the correct finite-witness cardinality in the pattern-color count.

theorem_23: Constructs a six-dimensional brick from six positive distances satisfying every squared triangle inequality.

theorem_24: Proves sphere product closure using finite witnesses and exact placement on every larger sphere.

theorem_25: Proves the few-color obstruction with the strict m less than ell endpoint and an explicit infinite-set reduction.

theorem_28: Reconstructs the two-stage finite-witness proof multiplying the allowed number of colors in a product.

theorem_5: Deduces the three-color assertion from the canonical seven-point spindle proof and records the seven-color counterexample as an external input.

theorem_6: Proves that every two-coloring of three-space contains a monochromatic unit equilateral triangle, with an exact parametrization of the torus step.

theorem_7: Derives a monochromatic unit square from a monochromatic four-cycle in a two-colored complete graph on six vertices and verifies the planar stripe counterexample.

theorem_8: Gives exact complex coordinates for the six congruent-triangle forcing gadget and handles collinear three-point sets without a genericity premise.

theorem_9: Forces an equilateral triangle at one of the scales d, sqrt(3)d, and 2d, then applies the planar six-triangle gadget.

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P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus, Euclidean Ramsey Theorems. I, Journal of Combinatorial Theory, Series A 14(3), May 1973, 341–363. Publisher record and DOI.

Source and scope

The copy read for this card is the published scan from the Rényi archive, with 23 physical pages matching printed pp. 341–363. The paper was received on 5 November 1972; this is distinct from publication in May 1973. The scan's later scanning metadata is not a mathematical revision date. The source record records this edition. The scan prints "Copyright © 1973 by Academic Press, Inc. All rights of reproduction in any form reserved" on its first page, every other right reserved.

The proof pages reconstruct the source's geometric, affine, field and product arguments, with their prerequisites and compressed steps expanded. The introductory Ramsey, van der Waerden and Gallai theorems, foundational compactness, and Theorem 5's negative seven-color construction are identified at their exact external scopes. The separate surface-embedding thesis comment is not treated as a complete proof. The external-input inventory and historical questions make these boundaries explicit.

Small configurations and explicit colorings

Theorem 5 distinguishes the positive three-color unit-pair statement from the external negative seven-color statement. The exact seven-point spindle has one canonical proof in the Paper II folder. The two-circle construction of Theorem 6 forces an equilateral triangle in three dimensions with two colors. Theorem 7 uses a monochromatic four-cycle to force a square, and Theorem 8 transfers a monochromatic equilateral triangle to any prescribed three-point configuration, including collinear triples.

Theorem 9 and Corollary 10 give the planar three-scale and 30–60–90 triangle deductions, and the paper then conjectures, in the unnumbered conjecture on p. 347, that every non-equilateral triangle is forced in two-colorings of the plane. Theorem 11 uses circles and a spherical band for the four-point L configuration. Theorem 12 records the distinct four-, three- and two-color shell obstructions for three, four and six equally spaced collinear points.

Spherical necessity and field colorings

The central Theorem 13 proves that a Ramsey set must be spherical, with a color count independent of ambient dimension. Its exact prerequisites are the affine-relation criterion, the real scalar obstruction, and the complete field-coloring theorem. The last proof includes the rational, transcendental, finite-algebraic and arbitrary-field cases, rather than referring out for its essential algebra.

Theorem 17 shows why the linear difference obstruction does not extend to arbitrary higher-degree forms. Corollary 18 gives an explicit $(2k)^k$ color bound under algebraic independence of the coefficient ratios; Corollary 19 handles irrational collinear triples with sixteen colors.

Products, bricks and few-color copies

The finite-witness reduction and product theorem prove Ramsey closure under orthogonal products. The brick corollaries give all brick subsets, including the regular-simplex centroid examples. Theorem 23 constructs four-point brick realizations from squared triangle inequalities; the five-point obstruction shows why the same condition is insufficient for larger configurations. Theorem 24 proves sphere-Ramsey bricks for every sufficiently large radius and dimension.

In the source's distinct $\ell$-Ramsey convention, a forced copy may use at most $\ell$ colors. The common-bisector criterion and paired-color separation prove Theorem 25: not fitting in $\ell-1$ concentric spheres precludes $m$-Ramsey for $m<\ell$. The finite polynomial witness supplies the compressed extension to infinite sets in a fixed Euclidean space. Theorem 28 multiplies the allowed color counts under finite orthogonal products.

Source precision and later connections

Theorem 5 says the seven-color pair-forcing statement is false, and Theorem 25 uses the strict endpoint $m<\ell$. Several source compressions also need care. Theorem 13 uses individual squared norms as colored scalar variables; Corollary 19's fractional errors are signed; the product theorem's pattern exponent is the finite witness size rather than its ambient dimension. The five-point nonobtuse example is affinely dependent, despite its informal “simplex” name. Every correction or expansion is explained on its result page and is distinguished from an author-issued erratum.

The original Ramsey product and spherical obstruction connect to the second paper, the Frankl–Rödl exponential simplex method, and the distinct recent proofs of Moore, Mirabi and Ivan–Leader–Walters. Historical questions in this paper are not silently promoted to current open-problem claims. No formal verification or Lean build is claimed.

Bears on.

• #174: Theorem 13 (p. 349) proves that every Ramsey set is spherical, a necessary condition; Theorem 20 (p. 357) and the brick corollaries (pp. 357–358) prove that products of Ramsey sets are Ramsey and that every subset of the vertices of a brick is Ramsey, so being such a subset is a sufficient condition (p. 360). The paper leaves open whether either condition is both necessary and sufficient (p. 360); it does not characterize the Ramsey sets.
• #173: Corollary 10 (p. 346) shows that the $30^\circ$–$60^\circ$ right triangle has a monochromatic copy in every two-coloring of the plane, and Theorem 9 (p. 346) that no such coloring misses the equilateral triangles of all three sides $d$, $\sqrt3d$, $2d$. The conjecture on p. 347 is posed there, unproved; its two clauses together imply the problem's statement. None of these decides the problem.

The paper's results concern copies in one color, or in few colors, with every color treated alike; none addresses the planar one-sided questions of Problems 188 and 214, in which one color class must avoid unit distance.

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