---

Source. Karamanlis, published p. 4, Lemma 4 (canonical PDF). The corresponding arXiv v1–v3 result is Lemma 3.2.

Karamanlis identifies this as a reformulation of Frankl–Pach–Reiher–Rödl, Borsuk and Ramsey type questions in Euclidean space, Lemma 4.9, and includes its proof. The proof reconstructed here is that included argument; it is not an independent review of the earlier chapter.

Statement. Let $n\ge2$ and let $(a_{ij})_{i,j=1}^{n}$ be a real symmetric array with $a_{ii}=0$ and $a_{ij}>0$ for $i\ne j$. Put $A=\max_{i<j}a_{ij}$. Suppose

$$\sum_{i<j}(A^2-a_{ij}^2)<A^2. \tag{1}$$

Then this array is the distance array of an affinely independent set of $n$ points in a product of at most $\binom n2$ regular simplices. Arrays and their realizations satisfying (1) are called almost regular.

Proof. Define

$$b^2=A^2-\sum_{i<j}(A^2-a_{ij}^2)>0, \qquad b_{ij}^2=A^2-a_{ij}^2\ge0.$$

Take a regular simplex $\Delta$ with $n$ vertices and edge length $b$. For each pair $i<j$ with $b_{ij}>0$, take a regular simplex $\Delta_{ij}$ with $n-1$ vertices and edge length $b_{ij}$. For $n=2$ there are no such pairs, because its sole distance is maximal. Thus no positive edge length is being assigned to a one-point factor.

Label the vertices of $\Delta$ by $[n]$. In the factor $\Delta_{ij}$, label its vertices by the $n-1$ classes of the partition of $[n]$ whose only nonsingleton class is $\{i,j\}$. For each label $s\in[n]$, let $z_s$ have base coordinate $s$ and, in each pair factor, the class containing $s$. This defines all $z_s$ simultaneously in

$$\Delta\times\prod_{i<j,\,b_{ij}>0}\Delta_{ij}.$$

For distinct labels $s,t$, the base coordinates are distinct. The coordinates in $\Delta_{ij}$ agree exactly when $\{s,t\}=\{i,j\}$. With $b_{st}=b_{ts}$ when needed, their squared distance is therefore

$$\|z_s-z_t\|^2 =b^2+\sum_{i<j}b_{ij}^2-b_{st}^2 =A^2-(A^2-a_{st}^2)=a_{st}^2.$$

Diagonal distances are zero directly. Projection onto the base factor sends the $n$ labels bijectively to the affinely independent vertices of $\Delta$. Any affine relation among the $z_s$ would project to one among those vertices, so all its coefficients vanish.

At least one pair attains $A$ and consequently has $b_{ij}=0$. There are at most $\binom n2-1$ nonzero pair factors; together with the base factor this gives at most $\binom n2$ factors. $\square$

Source precision. The printed distance calculation is introduced for all $s,t$; its displayed off-diagonal formula applies to $s\ne t$. The diagonal case is handled separately above. The partition labels also make explicit that no extra pair is identified in a pair factor. These are compilation explanations, not an author-issued erratum.

Use. Proposition 5 and Proposition 11.