---

Statement

The notions of a configuration and of an $E$-Ramsey configuration are on the definitions page. The paper quotes two results of Kříž, citing Theorems 3.2 and 4.1 of his 1991 paper (p. 3).

Product. If $F_1$ is $E_1$-Ramsey and $F_2$ is $E_2$-Ramsey, then $F_1\times F_2$ is $(E_1\times E_2)$-Ramsey; the same holds for finite products. The product relation is taken coordinatewise: two points are related when each pair of corresponding coordinates is related (p. 4).

Orbit gluing. Suppose $F$ is $E$-Ramsey and $b:F\to F$ is an isometry respecting $E$. For $z\in F$ and $r\ge1$, let $U(E;z,b,r)$ be the smallest equivalence relation containing $E$ in which $z,bz,\ldots,b^{r-1}z$ are equivalent. Then $F$ is $U(E;z,b,r)$-Ramsey.

The paper uses the product statement to pass from a configuration to its $(n+1)$-fold power, and orbit gluing only with $r=2$, both in the proof of Theorem 3.2. Kříž's own statements are recorded on his card as Theorem 3.2 and Theorem 4.1.

Source. Section 3, p. 3, with the coordinatewise product relation on p. 4, of Mostafa Mirabi, One-point extensions of Euclidean Ramsey sets, arXiv:2608.11736v1 (12 August 2026), the version named on the source card. The original is I. Kříž, Permutation groups in Euclidean Ramsey theory, Proc. Amer. Math. Soc. 112 (1991), no. 3, 899–907.

Read depth. Claims checked: Mirabi's statements were read clause by clause on the page image. Their proofs are Kříž's and are not part of this paper.

Proof pointer

Not proved in this paper; see the Kříž card.

Dependencies

None within this paper.

Bears on

• Problem 174: inputs to the proof of Theorem 3.2 and so of the closure property of Theorem 1.1.