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Claim. P. Frankl and V. Rödl, A partition property of simplices in Euclidean space, J. Amer. Math. Soc. 3 (1990), no. 1, 1--7, cited as [FrRo90] in the site's commentary on Problem 174. Theorem 5.1 (pp. 5--6) states that every finite affinely independent configuration, that is, the vertex set of a nondegenerate simplex, is super-Ramsey: for each such $A$ there are $\epsilon>0$ and finite sets $X_n\subseteq\mathbb R^n$ such that every subset of $X_n$ with more than $(1+\epsilon)^{-n}|X_n|$ points contains a congruent copy of $A$, as the paper's Definition 2.1 makes precise. Consequently every coloring of $\mathbb R^n$, $n$ large, with at most $(1+\epsilon)^n$ colors has a monochromatic congruent copy of $A$, and every nondegenerate simplex is Ramsey in the sense of Problem 174, the required dimension being $O_A(1+\log r)$ for $r$ colors. The proof contracts the simplex slightly, approximates the contracted squared distances by a near-regular brick subset built from Frankl and Wilson's two-point theorem and the authors' 1987 theorem on full joint-partition patterns, and restores the exact distances by an orthogonal product. The library's reconstructions are Theorem 5.1 and its Ramsey consequence; the card is frankl_1990_partition_property_simplices_euclidean_space.

Covers. The class of nondegenerate simplices: every finite affinely independent set is Ramsey, with the stronger density property. The paper does not decide affinely dependent sets; the full classification is OpenAI's accepted claim.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper is a journal publication in the Journal of the American Mathematical Society, volume 3, issue 1 (January 1990), the refereed evidence; the issue carries no day, so this page is dated to the first day of that month. The site's curator credits non-degenerate simplices to [FrRo90] in the problem's commentary, but the site labels the problem OPEN, so that credit is not reviewed evidence. The library's complete reconstruction of the same-paper chain is the corpus's own reading and is not an independent review.