Erdos 1975 euclidean ramsey theorems ii

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chromatic_translation_bridge: Gives the complete finite graph coloring argument behind the source’s chromatic observation.

corollary_6: Combines the exact density witness and counting transfer with the source floor convention.

definitions: Distinguishes congruent copies, translates, color order and historical statements.

external_inputs: Records the older monochromatic-triple theorem separately from the self-contained finite deductions.

grid_counterexample: Expands the square-array coloring and verifies avoidance for every translation and rotation of the grid.

later_source_inventory: Inventories the infinite-configuration and edge-coloring results without assigning them completed-proof status.

lattice_three_term_density: Checks the two residue-color constructions and the asymptotically sharp integer-grid density.

product_grid_lemma: Proves the complete inductive counting lemma that produces bricks from dense finite point sets.

seven_point_spindle: Supplies explicit coordinates and the independence bound behind the translation theorem.

theorem_1: Reconstructs the forced-color argument from an exact external monochromatic-triple input.

theorem_10: Gives the parity proof for any positive common edge length, including self-crossing polygons.

theorem_1_prime: Gives the complete two-circle proof of the historical planar four-point bound.

theorem_2: Proves the sphere-and-circles construction and its rectangle extension with an explicit dimensional limit.

theorem_3: Uses seven-point incidence counting to force a prescribed translate under red-distance exclusion.

theorem_4: Embeds the product-grid lemma in orthogonal coordinate blocks with the exact source exponents.

theorem_5: Proves the finite union-bound transfer from dense red obstructions to a prescribed blue configuration.

theorem_7: Reconstructs the edge-midpoint proof and separates its valid threshold from the printed n-point wording.

theorem_8: Corrects the circle equation explicitly and proves the counted family with a finite endpoint.

theorem_9: Checks the four-block construction, its eightfold counting multiplicity and the asymptotic threshold.

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Paul Erdős, Ronald L. Graham, Peter Montgomery, Bruce L. Rothschild, Joel Spencer, and Ernst G. Straus, Euclidean Ramsey Theorems, II, in Infinite and Finite Sets (Keszthely 1973), Colloquia Mathematica Societatis János Bolyai 10, North-Holland (1975), 529–557. MR 52 #2935; Zbl 313.05002.

Source copies and compilation scope

Two scans were read for this card. The first is the scan of the Rényi paper archive. The second, an alternate scan of unrecorded origin, is generally sharper and is used for the page references below. All 29 printed pages, 529–557, were visually compared between the scans; they contain the same visible article and page order. Their PDF encodings differ. No notice is printed in the archive scan; the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, prints "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only."); the colloquium volume has no publisher page, and no Crossref license is recorded; the term is unstated. No notice is printed in the alternate scan either; the term is unstated.

This compilation supplies 16 complete proof-bearing pages for the finite asymmetric, density and lattice arguments in Sections 2–3. Theorem 1’s triangular-lattice proof retains its exact external monochromatic-triple input. The later-section inventory records the remaining numbered results and their proof obligations; the whole paper is not labeled fully proved here. See the notation and scope conventions.

Asymmetric finite configurations

The paper gives two materially different proofs of the four-blue-point conclusion. Theorem 1 uses a forced-color triangular-lattice configuration after importing a monochromatic triple in three dimensions. Theorem 1′ works directly in the plane with two concentric circles. The latter establishes the historical four-point bound relevant to Problem 188, without claiming a five- or six-point result.

Theorem 2 forces a red right unit triangle or a blue unit square in three dimensions, with a rectangle extension. Its sphere-and-circles proof is not a solution of the planar Problem 214, which the paper leaves undecided.

The seven-point unit-distance configuration has independence number at most two, as shown with explicit coordinates. Theorem 3 applies it to three translated bad-choice sets and forces a blue translate of every prescribed three-point planar set when red distance $d>0$ is excluded. The chromatic translation argument gives the related graph-coloring obstruction.

The large-grid counterexample shows why arbitrary finite planar configurations cannot all be forced this way: a periodic array of red half-unit squares avoids red unit pairs but meets every congruent copy of a particular $10^{12}$-point grid. The complete proof checks arbitrary translations, rotations and boundaries. The red set is an array of squares, not strips.

Density and counting constructions

The product-grid lemma gives the full common-fiber counting induction. Theorem 4 embeds that product into orthogonal coordinate blocks and forces a prescribed brick in every sufficiently large subset. Theorem 5 converts any such finite density witness into an asymmetric translation conclusion. Corollary 6 combines them, with the exact dimension $m=n^{2^k}$ and the printed floor convention for the blue set's size.

Theorem 7's supported construction associates coordinate pairs to graph edges and uses the classification of graphs without a three-edge simple path. Its verified threshold is $n+1$ points, as in the printed proof. Theorem 8 places an orthogonal circle at unit distance from every vertex of a simplex and counts cubically many right unit triangles. Theorem 9 uses four disjoint coordinate blocks to obtain at least quadratically many unit squares.

The lattice density arguments give the asymptotically sharp two-thirds bound for avoiding unit three-term progressions in integer grids, and the corresponding triangular-grid construction. Theorem 10 excludes every odd equilateral closed polygon from the integer lattice through a common-power-of-two parity argument.

Source precision and remaining limits

Four finite-source issues are explicit in the proof pages. The product-grid definition omits the base $n$ in one displayed index range; the preceding coordinate blocks and the complete induction supply the intended part sizes. Theorem 7 prints $n$ in its statement but proves $n+1$ for the displayed construction. Theorem 8's circle equation omits $y^2$, which must be restored to give the stated unit distances. Theorem 9's intermediate binomial-ratio comparison has the wrong direction; the complete reconstruction estimates the actual product and proves the needed $e^{-1}$ limit directly.

The external-input page separates the older Paper I triple theorem from the self-contained deductions. Later topological, set-theoretic, infinite-dimensional and edge-coloring arguments remain in the qualified inventory. A comparison of source scans does not certify their mathematical claims. No Lean build, present-day status census or priority determination was performed as part of this source compilation.

Bears on. Problem 188, Problem 214.

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