---

Source. Published p. 221, Theorem 3.2 and the preceding external-input discussion. (canonical PDF).

Every finite box, meaning the vertex set of an orthogonal rectangular parallelotope, is hyper-Ramsey. Zero edge factors may be omitted; the zero-dimensional box is a singleton.

Exact external input. Every two-point configuration $\{u,v\}$ with $\|u-v\|=a>0$ is hyper-Ramsey: for each $\alpha>0$, there are $c>1$, $0<\epsilon<1$ and $m_0$ such that every $m\ge m_0$ has a finite nonempty witness on $S(\sqrt{a^2/4+\alpha},m)$ with size less than $c^m$ and with the weak forcing threshold $(1-\epsilon)^m$ of definitions. The source attributes this to Frankl–Wilson (1981), with further references to Graham (1983) and Rödl (1983). Those original proofs are external to this compilation.

Proof.

A box with positive edge lengths $a_1,\ldots,a_s$ is the orthogonal product of its $s$ two-point factors, and its intrinsic squared radius is $\sum_{i=1}^s a_i^2/4$. Given any $\alpha>0$, assign squared slack $\alpha/s$ to each factor. The quoted two-point theorem and repeated application of lemma_3_4 give the product with total squared slack $\alpha$. Since $\alpha$ was arbitrary, the box is hyper-Ramsey. The case $s=0$ is the singleton construction in definitions.

Source precision.

This conclusion concerns the whole box. A subset inherits the forcing property on the box's witness spheres, but the argument alone does not give arbitrary small slack above the subset's own smaller intrinsic circumradius. That distinction is respected in lemma_3_13. The 1990 product proof is separately preserved at theorem_6_4.

Bears on. #174.