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Source. Published p. 220, Lemma 2.3, with its set-up on pp. 219–220, explicitly imported from Matoušek–Rödl (1995). (canonical PDF).

For a unit vector $a=(a_1,\ldots,a_k)\in\mathbb R^k$ and an ordered set $K=\{u_1<\cdots<u_k\}\subseteq[s]$, define

$$\operatorname{spread}(a,K)=\sum_{j=1}^k a_je_{u_j}\in\mathbb R^s.$$

For every real $\eta>0$ and every integer $d$ (the application has $d\ge1$), there are integers $s$ and $k$, a unit vector $a\in\mathbb R^k$, and disjoint $k$-sets $\widetilde K_1<\cdots<\widetilde K_d$ in $[s]$, such that, for

$$Z=\operatorname{span}\{ \operatorname{spread}(a,\widetilde K_i):1\le i\le d\},$$

every unit vector $z\in Z$ is at distance at most $\eta$ from $\operatorname{spread}(a,K)$ for some $k$-set $K\subseteq[s]$. Here $I<J$ means every element of $I$ is smaller than every element of $J$. The disjoint supports and $\|a\|=1$ make the displayed spanning vectors orthonormal, so $Z$ has dimension exactly $d$, and the disjoint blocks force $s\ge dk$.

External proof scope. This is the exact lemma quoted by the canonical 2004 paper. Its non-elementary approximation proof is not included here. The original reference is J. Matoušek and V. Rödl, On Ramsey sets in spheres, Journal of Combinatorial Theory, Series A 70 (1995), 30–44, DOI 10.1016/0097-3165(95)90078-0. The present source compilation does not assert a full review of that original article. Neither distinctness nor nonvanishing of the individual coefficients $a_j$ is assumed; the later density transfer treats repeated values and zeros explicitly.

Bears on. #174.