Strong Ramsey properties of simplices

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circumradius_continuity: Proves intrinsic circumradius formulas, strict radius loss on affine projection, and continuity under small perturbations of a simplex.

claim_3_6: Checks the complete common-marginal and positive-cell properties of the constructed ordered partitions.

claim_3_7: Computes all pairwise intersections of the partition construction, including mixed zero-label cases.

definitions: Defines strong and hyper-Ramsey witnesses with precise radius, density and dimension conventions, and proves the required elementary transfers.

fact_3_10: Extends fixed-radius exponential-density witnesses from an arithmetic progression, or any bounded-gap dimension sequence, to all large dimensions.

lemma_2_3: Records the precise Matoušek–Rödl sphere approximation input, including ordered disjoint blocks and a common unit coefficient vector.

lemma_3_12: Proves a box realization for every near-regular squared-distance array, with at most one coordinate per pair and an explicit radius bound.

lemma_3_13: Proves the sufficient hyper-Ramsey slack bound with the correct squared-length scaling and an inclusive endpoint.

lemma_3_4: Proves the product theorem for specified squared slacks, including finite copy counting, singleton factors and every sufficiently large dimension.

lemma_3_5: Constructs linearly independent partition vectors with every joint cell positive and an explicit uniform squared-distance error.

lemma_3_9: Builds a fixed simplex close to any given simplex, with an exact controlled witness radius and an exponential density threshold.

remark_3_8: Proves that keeping the block-size ratio fixed preserves the complete target metric as the witness dimension grows.

theorem_1_3: Records the earlier qualitative near-circumsphere theorem for simplices as an external historical statement.

theorem_1_6: Derives exponential color forcing on every sphere with fixed positive radius slack above a simplex circumradius.

theorem_2_1: States the exact finite squared-distance criterion used in the source and links its complete elementary Gram proof.

theorem_2_2: States the imported dense-family partition theorem with all integer, marginal, positivity and uniformity hypotheses explicit.

theorem_3_2: Derives the hyper-Ramsey property of every finite box from the exact external two-point theorem and the fixed-slack product lemma.

theorem_3_3: Proves the complete simplex hyper-Ramsey theorem with a corrected near-regular radius budget and exact final spherical witnesses.

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Peter Frankl and Vojtěch Rödl, Strong Ramsey properties of simplices, Israel Journal of Mathematics 139 (2004), 215–236. DOI 10.1007/BF02787550. The canonical published PDF is the unchanged 22-page scan hosted on Frankl's author site.

The publisher record and printed first page identify volume 139 (2004). The publisher records December 2004 as the issue date. The manuscript was received 27 May 2001 and revised 16 January 2003. Physical PDF pages 1–22 are printed pages 215–236, without a cover. The PDF's 2007 scan-generation metadata is not a later mathematical version or publication date. Primary records were checked. No equivalence to a separate technical-report version is asserted. No notice is printed; the publisher's article page shows only the site-level notice "© 2026 Springer Nature", marks the article as a preview of subscription content and names no Creative Commons or open-access license (https://link.springer.com/article/10.1007/BF02787550, read 2026-10-02), every other right reserved.

Results and complete proof chain

The strongest result is theorem_3_3: every simplex is hyper-Ramsey. For each positive squared-radius slack $\alpha$, it gives finite witnesses on the sphere of radius $\sqrt{\rho(X)^2+\alpha}$, with exponential cardinality and an exponential density threshold, in every sufficiently large dimension. theorem_1_6, which the paper calls its main result (pp. 221 and 232), deduces the advertised strong Ramsey property: for each $\delta>0$, a sphere of radius $\rho(X)+\delta$ forces the simplex under exponentially many colors. Neither statement asserts forcing at zero radius slack.

The following source deductions have complete rewritten proofs here:

• lemma_3_4 proves the product theorem for fixed squared slacks. fact_3_10 supplies all eventual dimensions, including a bounded-gap extension used in the product proof.
• lemma_3_5, claim_3_6, and claim_3_7 construct full positive joint patterns, prove independence and compute distances. remark_3_8 ensures that the target metric is fixed as dimension grows.
• lemma_3_9 combines these patterns with the external spread-vector approximation to produce close simplices with controlled witness radii. Its constant-fiber argument handles repeated and zero coordinate values.
• lemma_3_12 supplies an elementary finite cut-vector proof of the near-regular box realization and its dimension bound for every number of vertices. lemma_3_13 proves the corrected sufficient radius budget.
• theorem_3_3 contracts squared distances, approximates the contracted simplex, realizes a near-regular residual, and takes a product diagonal. circumradius_continuity supplies the finite Gram and radius facts.
• theorem_3_2 proves the box deduction at its exact external two-point input. definitions proves the elementary slack, scaling, density and dimension transfers used throughout.

The argument differs from the 1990 ordinary super-Ramsey proof by preserving quantitative control of the containing sphere throughout approximation and the product. The 1990 proof and its source corrections remain separate.

Exact external inputs and historical scope

theorem_2_2 states the full prescribed-joint-pattern density theorem from Frankl–Rödl (1987), with all integer and marginal hypotheses. lemma_2_3 states the Matoušek–Rödl (1995) spread-vector sphere approximation. The proofs of those two external results are not included. theorem_3_2 states the spherical two-point hyper-Ramsey input attributed in the source to Frankl–Wilson (1981), with Graham (1983) and Rödl (1983) also cited. Its original proof remains external. The finite negative-type criterion theorem_2_1 links the complete elementary Gram proof already compiled.

The earlier qualitative theorem theorem_1_3 is an external historical statement. The introduction's chronology, chromatic-number bounds and open questions describe the paper's period; they are not a current-status audit. The paper's main same-paper deductions and the explicit local expansions are complete at the named inputs. This does not claim complete proofs of all papers in its bibliography, formal kernel verification, or a new mathematical solution.

Source calculations and radius qualifications

On p. 232, the source turns the squared-distance bound $\beta(1+\mu)$ into an edge-length bound and consequently writes an $O_d(\beta^2)$ squared-radius estimate. lemma_3_12 proves the correct $O_d(\beta)$ estimate, and lemma_3_13 uses it with an inclusive endpoint. The main proof chooses $\beta<\alpha/[2(d+1)^2]$ in place of the printed square-root bound. This smaller positive choice satisfies all other constraints and retains the source's method. The repairs are supplied by this compilation; no author-issued erratum is being claimed.

Other explicit expansions cover the unchanged zero diagonal in negative type, actual realization of residual distances, intrinsic circumradius continuity, normalized rational approximation, noninjective partition encodings, strictly positive slack, and final dimension changes. Source notes on the corresponding pages distinguish these from the printed text.

An arbitrary subconfiguration need not inherit hyper-Ramsey forcing at its own smaller intrinsic radius from the containing configuration. The proof here always tracks the actual sphere used by its witnesses. It does not fill the general intrinsic-radius subset clause left uncompiled in corollary_6_5.

Bears on. #174: for a simplex $X$ with circumradius $\rho$ and each $\alpha>0$, Theorem 3.3 gives constants $c>1$ and $0<\epsilon<1$ and, in every large dimension $m$, a finite set of fewer than $c^m$ points on the sphere of squared radius $\rho^2+\alpha$ in $\mathbb R^m$ in which every subset of relative size at least $(1-\epsilon)^m$ contains a congruent copy of $X$. For each $\delta>0$, Theorem 1.6 gives a $\sigma>0$ and a monochromatic congruent copy of $X$ in every colouring of the sphere of radius $\rho+\delta$ in $\mathbb R^n$ with at most $(1+\sigma)^n$ colours, for every large $n$. That every simplex is Ramsey was proved in 1990; these are sphere refinements for one class of Ramsey sets and do not characterise the Ramsey sets.

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