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Claim. Let $q$ be a prime power, $n\ge1$ an integer, $A_1,\ldots,A_n$ finite-dimensional vector spaces over $\mathbb F_q$, and $C$ a linear subspace of $A_1\oplus\cdots\oplus A_n$. A word $c=(c_1,\ldots,c_n)$ represents the $n$-element set $S(c)=\{(i,c_i):1\le i\le n\}$, and distinct words represent distinct sets. A $k$-sunflower consists of $k$ distinct sets whose pairwise intersections all equal one set, which may be empty. For every integer $k\ge3$ and every $a\in A_1\oplus\cdots\oplus A_n$, if the $n$-uniform family

$$\mathcal F=\{S(c):c\in a+C\}$$

contains no $k$-sunflower, then

$$\lvert\mathcal F\rvert=\lvert C\rvert<k^{2n}.$$

More precisely, $\dim C<n\lceil\log_qk\rceil$ and $\lvert C\rvert\le q^{n\lceil\log_qk\rceil-1}$; if $q\ge k$, then $C=\{0\}$ and $\mathcal F$ has one member. The bound comes from a subspace theorem: for every integer $s\ge1$, if $\dim C\ge sn$, there is an $s$-dimensional subspace $U$ of $C$ on which each coordinate projection is zero or injective, and for every $a$ the $q^s$ sets $S(c)$, $c\in a+U$, form a $q^s$-sunflower.

Covers. The special case of Problem 20 for affine linear word families: the $n$-uniform families $\{S(c):c\in a+C\}$ above, over any finite field $\mathbb F_q$ and with coordinate alphabets $A_i$ of any finite dimension. For every $k\ge3$, every such family with at least $k^{2n}$ members contains a $k$-sunflower, and the base $k^2$ does not depend on the field or on the alphabet dimensions. The result does not bound $f(n,k)$, which ranges over all $n$-uniform families: a family need not have this form, and passing from an arbitrary word family to its affine span can introduce new words and new sunflowers. Families outside the class are not addressed, and the general question is not settled by this claim.

Credit. The independent-representative criterion in the proof is the classical linear version of Hall's condition; the proof gives it in full and uses no external theorem. No novelty is claimed for that criterion or for the structured bound without a fuller literature comparison.

Standing. L268 is author-recorded at tier 0: a complete noncomputational argument, without independent examination or a Lean proof. Catalog acceptance is pending.

Depends on. L268 (tier 0), which consumes no other native claim and no external theorem.