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Source. Published p. 282, Theorem 9.1 and its preceding remark (PDF).

The source defines $mS+b$ on the same points as $S\subseteq H(t,q)$ by replacing each distance between distinct points by $m d+b$. It then asserts that if $m,b>\eta n$ and $mt+b<(1-\eta)n$, every sufficiently dense subset of $H(n,q)$ contains an isometric copy. No proof is given beyond the preceding positive-pattern discussion.

What the pattern theorem does prove. Given a dense family in a specified type class, and a compatible positive joint array for $r$ words with every cell at least a fixed positive proportion of $n$, Theorem 1.16 realizes that entire array. The pairwise distances are then obtained by summing the cells in which the corresponding two letter indices differ. This is the precise feasible-array consequence stated in the preceding source remark. The simple size conditions in Theorem 9.1 are not shown here to establish such feasibility.

Dependencies. theorem_1_16.