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Source. Published pp. 4–5, Lemma 4.1.

Statement. For every integer $s\ge2$ and $\epsilon>0$, there exists a super-Ramsey set $B=\{b_1,\ldots,b_s\}\subset\mathbb R^{s-1}$ such that

$$\left|\|b_i-b_j\|^2-(i-j)^2\right|<\epsilon\quad(i<j).$$

Proof relative to the joint-partition theorem. Shrink $\epsilon$ below $1/2$ if necessary. Choose an integer $t\ge s$ and, for $1\le i\le s$, let $x^{(i)}\in\mathbb R^{2t+s}$ be the shifted triangular word

$$x_j^{(i)}= \begin{cases} 0,&j<i\text{ or }j>2t+i,\\ j-i+1,&i\le j\le t+i,\\ 2t+1+i-j,&t+i<j\le2t+i. \end{cases}$$

All these words have the same counts of each symbol in $\{0,\ldots,t+1\}$. For $k=|i-j|\le s-1$, a direct finite calculation gives

$$\|x^{(i)}-x^{(j)}\|^2=2(t+1)k^2-k^3+k.$$

Here is that calculation. Put $T=t+1$ and $h(a)=\max(0,T-|a-T|)$ for integer $a$; the displayed words are translates of $h$ with their entire supports included. The difference $\Delta h(a)=h(a)-h(a-1)$ is $+1$ on $1,\ldots,T$, $-1$ on $T+1,\ldots,2T$, and zero elsewhere. For $0\le u\le T$ its shifted inner product is $2T-3u$. Since $h(a)-h(a-k)=\sum_{v=0}^{k-1}\Delta h(a-v)$, its squared norm is

$$2Tk+2\sum_{u=1}^{k-1}(k-u)(2T-3u)=2Tk^2-k^3+k.$$

The error relative to $2tk^2$ therefore has absolute value at most $k^3+2k^2+k=(k+1)^2k<4s^4$, as used in the source.

Let $q=t+2$ and form the $s$ by $q^s$ matrix whose columns list every word of length $s$ over $\{0,\ldots,t+1\}$ exactly once. Denote its rows by $y^{(i)}$. Each row has each symbol exactly $q^{s-1}$ times, and every joint pattern of the $s$ rows occurs once. For an integer $m>0$, set

$$z^{(i)}=\underbrace{x^{(i)}*\cdots*x^{(i)}}_{m\text{ copies}}*y^{(i)}, \qquad H=m(2t+s)+q^s.$$

The $z^{(i)}$ have equal symbol counts. Also $\|y^{(i)}-y^{(j)}\|^2\le q^s(t+1)^2$. Thus, for $\widetilde b_i=z^{(i)}/\sqrt{2tm}$,

$$\left|\|\widetilde b_i-\widetilde b_j\|^2-(i-j)^2\right| <\frac{4s^4}{2t}+\frac{q^s(t+1)^2}{2tm}.$$

First fix $t$ so the first term is less than $\epsilon/2$, and then fix $m$ so the second term is less than $\epsilon/2$. From now on $t,m,H$ and this single target configuration are fixed. Its points are distinct, and its affine span has dimension at most $s-1$.

For each positive integer $p$, concatenate $p$ copies of each $z^{(i)}$ to obtain $\omega^{(i)}\in\mathbb R^n$, where $n=pH$. The normalized configurations $\{\omega^{(i)}/\sqrt{2tmp}:1\le i\le s\}$ are all congruent to the fixed $\{\widetilde b_i\}$: their squared distances are independent of $p$. This prevents the witness dimension from changing the theorem's target.

Partition $[n]$ according to the symbol values in each $\omega^{(i)}$. Let $l_a$ be the common count of symbol $a$, and let $M$ be their full $s$-fold joint-intersection array. Each joint cell occurs at least $p$ times, from the repeated $y$ blocks, so $m_{\mathbf j}\ge p=n/H$. Every $l_a$ is positive, and every one-coordinate marginal is $(l_a)_a$. Apply the precise joint-partition input with $r=s$, alphabet size $q$, and fixed $\eta=1/H$.

Take $X_n$ to be all words of length $n$ having these symbol counts, scaled by $1/\sqrt{2tmp}$. Its cardinality is $n!/\prod_a l_a!<q^n$. A subset of size at least $(1-\epsilon')^n|X_n|$, for the fixed constant $\epsilon'>0$ supplied by that theorem, contains $s$ words with the prescribed full joint array. That array determines each pairwise squared distance by summing $(a-b)^2$ over the corresponding marginal cells. After scaling, the resulting configuration is congruent to the fixed target. Therefore every avoiding subset has size less than $(1-\epsilon')^n|X_n|$.

The dimension extension from multiples of $H$ to every sufficiently large ambient dimension gives the super-Ramsey witnesses. Finally, identify the target's affine span isometrically with a subspace of $\mathbb R^{s-1}$ to obtain $B$.

Source precision. The source calls $y^{(i)}$ a column, but its declared length $q^s$, equal marginals and pattern construction require the $i$th row. Its later pair-index upper bound $m$ is $s$ for these $s$ points. The proof above also spells out the fixed normalized target and the all-dimension step. These are compilation explanations and corrections, not an author erratum.

Proof scope. Complete relative to the exact external joint-partition theorem; no proof of that 1987 theorem is claimed here.

Bears on. #174.